import random
import warnings

from sympy import (
    Abs, Add, E, Float, I, Integer, Max, Min, N, Poly, Pow, PurePoly, Rational,
    S, Symbol, cos, exp, expand_mul, oo, pi, signsimp, simplify, sin, sqrt, symbols,
    sympify, trigsimp, tan, sstr, diff, Function)
from sympy.matrices.matrices import (ShapeError, MatrixError,
    NonSquareMatrixError, DeferredVector, _find_reasonable_pivot_naive,
    _simplify)
from sympy.matrices import (
    GramSchmidt, ImmutableMatrix, ImmutableSparseMatrix, Matrix,
    SparseMatrix, casoratian, diag, eye, hessian,
    matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
    rot_axis3, wronskian, zeros, MutableDenseMatrix, ImmutableDenseMatrix)
from sympy.core.compatibility import long, iterable, range, Hashable
from sympy.core import Tuple
from sympy.utilities.iterables import flatten, capture
from sympy.utilities.pytest import raises, XFAIL, slow, skip
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.solvers import solve
from sympy.assumptions import Q
from sympy.tensor.array import Array

from sympy.abc import a, b, c, d, x, y, z, t

# don't re-order this list
classes = (Matrix, SparseMatrix, ImmutableMatrix, ImmutableSparseMatrix)


def test_args():
    for c, cls in enumerate(classes):
        m = cls.zeros(3, 2)
        # all should give back the same type of arguments, e.g. ints for shape
        assert m.shape == (3, 2) and all(type(i) is int for i in m.shape)
        assert m.rows == 3 and type(m.rows) is int
        assert m.cols == 2 and type(m.cols) is int
        if not c % 2:
            assert type(m._mat) in (list, tuple, Tuple)
        else:
            assert type(m._smat) is dict


def test_division():
    v = Matrix(1, 2, [x, y])
    assert v.__div__(z) == Matrix(1, 2, [x/z, y/z])
    assert v.__truediv__(z) == Matrix(1, 2, [x/z, y/z])
    assert v/z == Matrix(1, 2, [x/z, y/z])


def test_sum():
    m = Matrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
    assert m + m == Matrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
    n = Matrix(1, 2, [1, 2])
    raises(ShapeError, lambda: m + n)

def test_abs():
    m = Matrix(1, 2, [-3, x])
    n = Matrix(1, 2, [3, Abs(x)])
    assert abs(m) == n

def test_addition():
    a = Matrix((
        (1, 2),
        (3, 1),
    ))

    b = Matrix((
        (1, 2),
        (3, 0),
    ))

    assert a + b == a.add(b) == Matrix([[2, 4], [6, 1]])


def test_fancy_index_matrix():
    for M in (Matrix, SparseMatrix):
        a = M(3, 3, range(9))
        assert a == a[:, :]
        assert a[1, :] == Matrix(1, 3, [3, 4, 5])
        assert a[:, 1] == Matrix([1, 4, 7])
        assert a[[0, 1], :] == Matrix([[0, 1, 2], [3, 4, 5]])
        assert a[[0, 1], 2] == a[[0, 1], [2]]
        assert a[2, [0, 1]] == a[[2], [0, 1]]
        assert a[:, [0, 1]] == Matrix([[0, 1], [3, 4], [6, 7]])
        assert a[0, 0] == 0
        assert a[0:2, :] == Matrix([[0, 1, 2], [3, 4, 5]])
        assert a[:, 0:2] == Matrix([[0, 1], [3, 4], [6, 7]])
        assert a[::2, 1] == a[[0, 2], 1]
        assert a[1, ::2] == a[1, [0, 2]]
        a = M(3, 3, range(9))
        assert a[[0, 2, 1, 2, 1], :] == Matrix([
            [0, 1, 2],
            [6, 7, 8],
            [3, 4, 5],
            [6, 7, 8],
            [3, 4, 5]])
        assert a[:, [0,2,1,2,1]] == Matrix([
            [0, 2, 1, 2, 1],
            [3, 5, 4, 5, 4],
            [6, 8, 7, 8, 7]])

    a = SparseMatrix.zeros(3)
    a[1, 2] = 2
    a[0, 1] = 3
    a[2, 0] = 4
    assert a.extract([1, 1], [2]) == Matrix([
    [2],
    [2]])
    assert a.extract([1, 0], [2, 2, 2]) == Matrix([
    [2, 2, 2],
    [0, 0, 0]])
    assert a.extract([1, 0, 1, 2], [2, 0, 1, 0]) == Matrix([
        [2, 0, 0, 0],
        [0, 0, 3, 0],
        [2, 0, 0, 0],
        [0, 4, 0, 4]])


def test_multiplication():
    a = Matrix((
        (1, 2),
        (3, 1),
        (0, 6),
    ))

    b = Matrix((
        (1, 2),
        (3, 0),
    ))

    c = a*b
    assert c[0, 0] == 7
    assert c[0, 1] == 2
    assert c[1, 0] == 6
    assert c[1, 1] == 6
    assert c[2, 0] == 18
    assert c[2, 1] == 0

    try:
        eval('c = a @ b')
    except SyntaxError:
        pass
    else:
        assert c[0, 0] == 7
        assert c[0, 1] == 2
        assert c[1, 0] == 6
        assert c[1, 1] == 6
        assert c[2, 0] == 18
        assert c[2, 1] == 0

    h = matrix_multiply_elementwise(a, c)
    assert h == a.multiply_elementwise(c)
    assert h[0, 0] == 7
    assert h[0, 1] == 4
    assert h[1, 0] == 18
    assert h[1, 1] == 6
    assert h[2, 0] == 0
    assert h[2, 1] == 0
    raises(ShapeError, lambda: matrix_multiply_elementwise(a, b))

    c = b * Symbol("x")
    assert isinstance(c, Matrix)
    assert c[0, 0] == x
    assert c[0, 1] == 2*x
    assert c[1, 0] == 3*x
    assert c[1, 1] == 0

    c2 = x * b
    assert c == c2

    c = 5 * b
    assert isinstance(c, Matrix)
    assert c[0, 0] == 5
    assert c[0, 1] == 2*5
    assert c[1, 0] == 3*5
    assert c[1, 1] == 0

    try:
        eval('c = 5 @ b')
    except SyntaxError:
        pass
    else:
        assert isinstance(c, Matrix)
        assert c[0, 0] == 5
        assert c[0, 1] == 2*5
        assert c[1, 0] == 3*5
        assert c[1, 1] == 0


def test_power():
    raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)

    R = Rational
    A = Matrix([[2, 3], [4, 5]])
    assert (A**-3)[:] == [R(-269)/8, R(153)/8, R(51)/2, R(-29)/2]
    assert (A**5)[:] == [6140, 8097, 10796, 14237]
    A = Matrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
    assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
    assert A**0 == eye(3)
    assert A**1 == A
    assert (Matrix([[2]]) ** 100)[0, 0] == 2**100
    assert eye(2)**10000000 == eye(2)
    assert Matrix([[1, 2], [3, 4]])**Integer(2) == Matrix([[7, 10], [15, 22]])

    A = Matrix([[33, 24], [48, 57]])
    assert (A**(S(1)/2))[:] == [5, 2, 4, 7]
    A = Matrix([[0, 4], [-1, 5]])
    assert (A**(S(1)/2))**2 == A

    assert Matrix([[1, 0], [1, 1]])**(S(1)/2) == Matrix([[1, 0], [S.Half, 1]])
    assert Matrix([[1, 0], [1, 1]])**0.5 == Matrix([[1.0, 0], [0.5, 1.0]])
    from sympy.abc import a, b, n
    assert Matrix([[1, a], [0, 1]])**n == Matrix([[1, a*n], [0, 1]])
    assert Matrix([[b, a], [0, b]])**n == Matrix([[b**n, a*b**(n-1)*n], [0, b**n]])
    assert Matrix([[a, 1, 0], [0, a, 1], [0, 0, a]])**n == Matrix([
        [a**n, a**(n-1)*n, a**(n-2)*(n-1)*n/2],
        [0, a**n, a**(n-1)*n],
        [0, 0, a**n]])
    assert Matrix([[a, 1, 0], [0, a, 0], [0, 0, b]])**n == Matrix([
        [a**n, a**(n-1)*n, 0],
        [0, a**n, 0],
        [0, 0, b**n]])

    A = Matrix([[1, 0], [1, 7]])
    assert A._matrix_pow_by_jordan_blocks(3) == A._eval_pow_by_recursion(3)
    A = Matrix([[2]])
    assert A**10 == Matrix([[2**10]]) == A._matrix_pow_by_jordan_blocks(10) == \
        A._eval_pow_by_recursion(10)

    # testing a matrix that cannot be jordan blocked issue 11766
    m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
    raises(MatrixError, lambda: m._matrix_pow_by_jordan_blocks(10))

    # test issue 11964
    raises(ValueError, lambda: Matrix([[1, 1], [3, 3]])._matrix_pow_by_jordan_blocks(-10))
    A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 0]])  # Nilpotent jordan block size 3
    assert A**10.0 == Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
    raises(ValueError, lambda: A**2.1)
    raises(ValueError, lambda: A**(S(3)/2))
    A = Matrix([[8, 1], [3, 2]])
    assert A**10.0 == Matrix([[1760744107, 272388050], [817164150, 126415807]])
    A = Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])  # Nilpotent jordan block size 1
    assert A**10.2 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
    A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 1]])  # Nilpotent jordan block size 2
    assert A**10.0 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
    n = Symbol('n', integer=True)
    raises(ValueError, lambda: A**n)
    n = Symbol('n', integer=True, nonnegative=True)
    raises(ValueError, lambda: A**n)
    assert A**(n + 2) == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
    raises(ValueError, lambda: A**(S(3)/2))
    A = Matrix([[0, 0, 1], [3, 0, 1], [4, 3, 1]])
    assert A**5.0 == Matrix([[168,  72,  89], [291, 144, 161], [572, 267, 329]])
    assert A**5.0 == A**5


def test_creation():
    raises(ValueError, lambda: Matrix(5, 5, range(20)))
    raises(ValueError, lambda: Matrix(5, -1, []))
    raises(IndexError, lambda: Matrix((1, 2))[2])
    with raises(IndexError):
        Matrix((1, 2))[1:2] = 5
    with raises(IndexError):
        Matrix((1, 2))[3] = 5

    assert Matrix() == Matrix([]) == Matrix([[]]) == Matrix(0, 0, [])

    a = Matrix([[x, 0], [0, 0]])
    m = a
    assert m.cols == m.rows
    assert m.cols == 2
    assert m[:] == [x, 0, 0, 0]

    b = Matrix(2, 2, [x, 0, 0, 0])
    m = b
    assert m.cols == m.rows
    assert m.cols == 2
    assert m[:] == [x, 0, 0, 0]

    assert a == b

    assert Matrix(b) == b

    c = Matrix((
        Matrix((
            (1, 2, 3),
            (4, 5, 6)
        )),
        (7, 8, 9)
    ))
    assert c.cols == 3
    assert c.rows == 3
    assert c[:] == [1, 2, 3, 4, 5, 6, 7, 8, 9]

    assert Matrix(eye(2)) == eye(2)
    assert ImmutableMatrix(ImmutableMatrix(eye(2))) == ImmutableMatrix(eye(2))
    assert ImmutableMatrix(c) == c.as_immutable()
    assert Matrix(ImmutableMatrix(c)) == ImmutableMatrix(c).as_mutable()

    assert c is not Matrix(c)


def test_tolist():
    lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
    m = Matrix(lst)
    assert m.tolist() == lst


def test_as_mutable():
    assert zeros(0, 3).as_mutable() == zeros(0, 3)
    assert zeros(0, 3).as_immutable() == ImmutableMatrix(zeros(0, 3))
    assert zeros(3, 0).as_immutable() == ImmutableMatrix(zeros(3, 0))


def test_determinant():

    for M in [Matrix(), Matrix([[1]])]:
        assert (
            M.det() ==
            M._eval_det_bareiss() ==
            M._eval_det_berkowitz() ==
            M._eval_det_lu() ==
            1)

    M = Matrix(( (-3,  2),
                 ( 8, -5) ))

    assert M.det(method="bareiss") == -1
    assert M.det(method="berkowitz") == -1

    M = Matrix(( (x,   1),
                 (y, 2*y) ))

    assert M.det(method="bareiss") == 2*x*y - y
    assert M.det(method="berkowitz") == 2*x*y - y

    M = Matrix(( (1, 1, 1),
                 (1, 2, 3),
                 (1, 3, 6) ))

    assert M.det(method="bareiss") == 1
    assert M.det(method="berkowitz") == 1

    M = Matrix(( ( 3, -2,  0, 5),
                 (-2,  1, -2, 2),
                 ( 0, -2,  5, 0),
                 ( 5,  0,  3, 4) ))

    assert M.det(method="bareiss") == -289
    assert M.det(method="berkowitz") == -289

    M = Matrix(( ( 1,  2,  3,  4),
                 ( 5,  6,  7,  8),
                 ( 9, 10, 11, 12),
                 (13, 14, 15, 16) ))

    assert M.det(method="bareiss") == 0
    assert M.det(method="berkowitz") == 0

    M = Matrix(( (3, 2, 0, 0, 0),
                 (0, 3, 2, 0, 0),
                 (0, 0, 3, 2, 0),
                 (0, 0, 0, 3, 2),
                 (2, 0, 0, 0, 3) ))

    assert M.det(method="bareiss") == 275
    assert M.det(method="berkowitz") == 275

    M = Matrix(( (1, 0,  1,  2, 12),
                 (2, 0,  1,  1,  4),
                 (2, 1,  1, -1,  3),
                 (3, 2, -1,  1,  8),
                 (1, 1,  1,  0,  6) ))

    assert M.det(method="bareiss") == -55
    assert M.det(method="berkowitz") == -55

    M = Matrix(( (-5,  2,  3,  4,  5),
                 ( 1, -4,  3,  4,  5),
                 ( 1,  2, -3,  4,  5),
                 ( 1,  2,  3, -2,  5),
                 ( 1,  2,  3,  4, -1) ))

    assert M.det(method="bareiss") == 11664
    assert M.det(method="berkowitz") == 11664

    M = Matrix(( ( 2,  7, -1, 3, 2),
                 ( 0,  0,  1, 0, 1),
                 (-2,  0,  7, 0, 2),
                 (-3, -2,  4, 5, 3),
                 ( 1,  0,  0, 0, 1) ))

    assert M.det(method="bareiss") == 123
    assert M.det(method="berkowitz") == 123

    M = Matrix(( (x, y, z),
                 (1, 0, 0),
                 (y, z, x) ))

    assert M.det(method="bareiss") == z**2 - x*y
    assert M.det(method="berkowitz") == z**2 - x*y

    # issue 13835
    a = symbols('a')
    M = lambda n: Matrix([[i + a*j for i in range(n)]
                          for j in range(n)])
    assert M(5).det() == 0
    assert M(6).det() == 0
    assert M(7).det() == 0


def test_det_LU_decomposition():

    for M in [Matrix(), Matrix([[1]])]:
        assert M.det(method="lu") == 1

    M = Matrix(( (-3,  2),
                 ( 8, -5) ))

    assert M.det(method="lu") == -1

    M = Matrix(( (x,   1),
                 (y, 2*y) ))

    assert M.det(method="lu") == 2*x*y - y

    M = Matrix(( (1, 1, 1),
                 (1, 2, 3),
                 (1, 3, 6) ))

    assert M.det(method="lu") == 1

    M = Matrix(( ( 3, -2,  0, 5),
                 (-2,  1, -2, 2),
                 ( 0, -2,  5, 0),
                 ( 5,  0,  3, 4) ))

    assert M.det(method="lu") == -289

    M = Matrix(( (3, 2, 0, 0, 0),
                 (0, 3, 2, 0, 0),
                 (0, 0, 3, 2, 0),
                 (0, 0, 0, 3, 2),
                 (2, 0, 0, 0, 3) ))

    assert M.det(method="lu") == 275

    M = Matrix(( (1, 0,  1,  2, 12),
                 (2, 0,  1,  1,  4),
                 (2, 1,  1, -1,  3),
                 (3, 2, -1,  1,  8),
                 (1, 1,  1,  0,  6) ))

    assert M.det(method="lu") == -55

    M = Matrix(( (-5,  2,  3,  4,  5),
                 ( 1, -4,  3,  4,  5),
                 ( 1,  2, -3,  4,  5),
                 ( 1,  2,  3, -2,  5),
                 ( 1,  2,  3,  4, -1) ))

    assert M.det(method="lu") == 11664

    M = Matrix(( ( 2,  7, -1, 3, 2),
                 ( 0,  0,  1, 0, 1),
                 (-2,  0,  7, 0, 2),
                 (-3, -2,  4, 5, 3),
                 ( 1,  0,  0, 0, 1) ))

    assert M.det(method="lu") == 123

    M = Matrix(( (x, y, z),
                 (1, 0, 0),
                 (y, z, x) ))

    assert M.det(method="lu") == z**2 - x*y


def test_slicing():
    m0 = eye(4)
    assert m0[:3, :3] == eye(3)
    assert m0[2:4, 0:2] == zeros(2)

    m1 = Matrix(3, 3, lambda i, j: i + j)
    assert m1[0, :] == Matrix(1, 3, (0, 1, 2))
    assert m1[1:3, 1] == Matrix(2, 1, (2, 3))

    m2 = Matrix([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]])
    assert m2[:, -1] == Matrix(4, 1, [3, 7, 11, 15])
    assert m2[-2:, :] == Matrix([[8, 9, 10, 11], [12, 13, 14, 15]])


def test_submatrix_assignment():
    m = zeros(4)
    m[2:4, 2:4] = eye(2)
    assert m == Matrix(((0, 0, 0, 0),
                        (0, 0, 0, 0),
                        (0, 0, 1, 0),
                        (0, 0, 0, 1)))
    m[:2, :2] = eye(2)
    assert m == eye(4)
    m[:, 0] = Matrix(4, 1, (1, 2, 3, 4))
    assert m == Matrix(((1, 0, 0, 0),
                        (2, 1, 0, 0),
                        (3, 0, 1, 0),
                        (4, 0, 0, 1)))
    m[:, :] = zeros(4)
    assert m == zeros(4)
    m[:, :] = [(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)]
    assert m == Matrix(((1, 2, 3, 4),
                        (5, 6, 7, 8),
                        (9, 10, 11, 12),
                        (13, 14, 15, 16)))
    m[:2, 0] = [0, 0]
    assert m == Matrix(((0, 2, 3, 4),
                        (0, 6, 7, 8),
                        (9, 10, 11, 12),
                        (13, 14, 15, 16)))


def test_extract():
    m = Matrix(4, 3, lambda i, j: i*3 + j)
    assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
    assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
    assert m.extract(range(4), range(3)) == m
    raises(IndexError, lambda: m.extract([4], [0]))
    raises(IndexError, lambda: m.extract([0], [3]))


def test_reshape():
    m0 = eye(3)
    assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
    m1 = Matrix(3, 4, lambda i, j: i + j)
    assert m1.reshape(
        4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
    assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))


def test_applyfunc():
    m0 = eye(3)
    assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
    assert m0.applyfunc(lambda x: 0) == zeros(3)


def test_expand():
    m0 = Matrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
    # Test if expand() returns a matrix
    m1 = m0.expand()
    assert m1 == Matrix(
        [[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])

    a = Symbol('a', real=True)

    assert Matrix([exp(I*a)]).expand(complex=True) == \
        Matrix([cos(a) + I*sin(a)])

    assert Matrix([[0, 1, 2], [0, 0, -1], [0, 0, 0]]).exp() == Matrix([
        [1, 1, Rational(3, 2)],
        [0, 1, -1],
        [0, 0, 1]]
    )

def test_refine():
    m0 = Matrix([[Abs(x)**2, sqrt(x**2)],
                [sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
    m1 = m0.refine(Q.real(x) & Q.real(y))
    assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])

    m1 = m0.refine(Q.positive(x) & Q.positive(y))
    assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])

    m1 = m0.refine(Q.negative(x) & Q.negative(y))
    assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])

def test_random():
    M = randMatrix(3, 3)
    M = randMatrix(3, 3, seed=3)
    assert M == randMatrix(3, 3, seed=3)

    M = randMatrix(3, 4, 0, 150)
    M = randMatrix(3, seed=4, symmetric=True)
    assert M == randMatrix(3, seed=4, symmetric=True)

    S = M.copy()
    S.simplify()
    assert S == M  # doesn't fail when elements are Numbers, not int

    rng = random.Random(4)
    assert M == randMatrix(3, symmetric=True, prng=rng)

    # Ensure symmetry
    for size in (10, 11): # Test odd and even
        for percent in (100, 70, 30):
            M = randMatrix(size, symmetric=True, percent=percent, prng=rng)
            assert M == M.T

    M = randMatrix(10, min=1, percent=70)
    zero_count = 0
    for i in range(M.shape[0]):
        for j in range(M.shape[1]):
            if M[i, j] == 0:
                zero_count += 1
    assert zero_count == 30


def test_LUdecomp():
    testmat = Matrix([[0, 2, 5, 3],
                      [3, 3, 7, 4],
                      [8, 4, 0, 2],
                      [-2, 6, 3, 4]])
    L, U, p = testmat.LUdecomposition()
    assert L.is_lower
    assert U.is_upper
    assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)

    testmat = Matrix([[6, -2, 7, 4],
                      [0, 3, 6, 7],
                      [1, -2, 7, 4],
                      [-9, 2, 6, 3]])
    L, U, p = testmat.LUdecomposition()
    assert L.is_lower
    assert U.is_upper
    assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)

    # non-square
    testmat = Matrix([[1, 2, 3],
                      [4, 5, 6],
                      [7, 8, 9],
                      [10, 11, 12]])
    L, U, p = testmat.LUdecomposition(rankcheck=False)
    assert L.is_lower
    assert U.is_upper
    assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)

    # square and singular
    testmat = Matrix([[1, 2, 3],
                      [2, 4, 6],
                      [4, 5, 6]])
    L, U, p = testmat.LUdecomposition(rankcheck=False)
    assert L.is_lower
    assert U.is_upper
    assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)

    M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
    L, U, p = M.LUdecomposition()
    assert L.is_lower
    assert U.is_upper
    assert (L*U).permute_rows(p, 'backward') - M == zeros(3)

    mL = Matrix((
        (1, 0, 0),
        (2, 3, 0),
    ))
    assert mL.is_lower is True
    assert mL.is_upper is False
    mU = Matrix((
        (1, 2, 3),
        (0, 4, 5),
    ))
    assert mU.is_lower is False
    assert mU.is_upper is True

    # test FF LUdecomp
    M = Matrix([[1, 3, 3],
                [3, 2, 6],
                [3, 2, 2]])
    P, L, Dee, U = M.LUdecompositionFF()
    assert P*M == L*Dee.inv()*U

    M = Matrix([[1,  2, 3,  4],
                [3, -1, 2,  3],
                [3,  1, 3, -2],
                [6, -1, 0,  2]])
    P, L, Dee, U = M.LUdecompositionFF()
    assert P*M == L*Dee.inv()*U

    M = Matrix([[0, 0, 1],
                [2, 3, 0],
                [3, 1, 4]])
    P, L, Dee, U = M.LUdecompositionFF()
    assert P*M == L*Dee.inv()*U


def test_LUsolve():
    A = Matrix([[2, 3, 5],
                [3, 6, 2],
                [8, 3, 6]])
    x = Matrix(3, 1, [3, 7, 5])
    b = A*x
    soln = A.LUsolve(b)
    assert soln == x
    A = Matrix([[0, -1, 2],
                [5, 10, 7],
                [8,  3, 4]])
    x = Matrix(3, 1, [-1, 2, 5])
    b = A*x
    soln = A.LUsolve(b)
    assert soln == x


def test_QRsolve():
    A = Matrix([[2, 3, 5],
                [3, 6, 2],
                [8, 3, 6]])
    x = Matrix(3, 1, [3, 7, 5])
    b = A*x
    soln = A.QRsolve(b)
    assert soln == x
    x = Matrix([[1, 2], [3, 4], [5, 6]])
    b = A*x
    soln = A.QRsolve(b)
    assert soln == x

    A = Matrix([[0, -1, 2],
                [5, 10, 7],
                [8,  3, 4]])
    x = Matrix(3, 1, [-1, 2, 5])
    b = A*x
    soln = A.QRsolve(b)
    assert soln == x
    x = Matrix([[7, 8], [9, 10], [11, 12]])
    b = A*x
    soln = A.QRsolve(b)
    assert soln == x


def test_inverse():
    A = eye(4)
    assert A.inv() == eye(4)
    assert A.inv(method="LU") == eye(4)
    assert A.inv(method="ADJ") == eye(4)
    A = Matrix([[2, 3, 5],
                [3, 6, 2],
                [8, 3, 6]])
    Ainv = A.inv()
    assert A*Ainv == eye(3)
    assert A.inv(method="LU") == Ainv
    assert A.inv(method="ADJ") == Ainv

    # test that immutability is not a problem
    cls = ImmutableMatrix
    m = cls([[48, 49, 31],
             [ 9, 71, 94],
             [59, 28, 65]])
    assert all(type(m.inv(s)) is cls for s in 'GE ADJ LU'.split())
    cls = ImmutableSparseMatrix
    m = cls([[48, 49, 31],
             [ 9, 71, 94],
             [59, 28, 65]])
    assert all(type(m.inv(s)) is cls for s in 'CH LDL'.split())


def test_matrix_inverse_mod():
    A = Matrix(2, 1, [1, 0])
    raises(NonSquareMatrixError, lambda: A.inv_mod(2))
    A = Matrix(2, 2, [1, 0, 0, 0])
    raises(ValueError, lambda: A.inv_mod(2))
    A = Matrix(2, 2, [1, 2, 3, 4])
    Ai = Matrix(2, 2, [1, 1, 0, 1])
    assert A.inv_mod(3) == Ai
    A = Matrix(2, 2, [1, 0, 0, 1])
    assert A.inv_mod(2) == A
    A = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
    raises(ValueError, lambda: A.inv_mod(5))
    A = Matrix(3, 3, [5, 1, 3, 2, 6, 0, 2, 1, 1])
    Ai = Matrix(3, 3, [6, 8, 0, 1, 5, 6, 5, 6, 4])
    assert A.inv_mod(9) == Ai
    A = Matrix(3, 3, [1, 6, -3, 4, 1, -5, 3, -5, 5])
    Ai = Matrix(3, 3, [4, 3, 3, 1, 2, 5, 1, 5, 1])
    assert A.inv_mod(6) == Ai
    A = Matrix(3, 3, [1, 6, 1, 4, 1, 5, 3, 2, 5])
    Ai = Matrix(3, 3, [6, 0, 3, 6, 6, 4, 1, 6, 1])
    assert A.inv_mod(7) == Ai


def test_util():
    R = Rational

    v1 = Matrix(1, 3, [1, 2, 3])
    v2 = Matrix(1, 3, [3, 4, 5])
    assert v1.norm() == sqrt(14)
    assert v1.project(v2) == Matrix(1, 3, [R(39)/25, R(52)/25, R(13)/5])
    assert Matrix.zeros(1, 2) == Matrix(1, 2, [0, 0])
    assert ones(1, 2) == Matrix(1, 2, [1, 1])
    assert v1.copy() == v1
    # cofactor
    assert eye(3) == eye(3).cofactor_matrix()
    test = Matrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
    assert test.cofactor_matrix() == \
        Matrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
    test = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    assert test.cofactor_matrix() == \
        Matrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])


def test_jacobian_hessian():
    L = Matrix(1, 2, [x**2*y, 2*y**2 + x*y])
    syms = [x, y]
    assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])

    L = Matrix(1, 2, [x, x**2*y**3])
    assert L.jacobian(syms) == Matrix([[1, 0], [2*x*y**3, x**2*3*y**2]])

    f = x**2*y
    syms = [x, y]
    assert hessian(f, syms) == Matrix([[2*y, 2*x], [2*x, 0]])

    f = x**2*y**3
    assert hessian(f, syms) == \
        Matrix([[2*y**3, 6*x*y**2], [6*x*y**2, 6*x**2*y]])

    f = z + x*y**2
    g = x**2 + 2*y**3
    ans = Matrix([[0,   2*y],
                  [2*y, 2*x]])
    assert ans == hessian(f, Matrix([x, y]))
    assert ans == hessian(f, Matrix([x, y]).T)
    assert hessian(f, (y, x), [g]) == Matrix([
        [     0, 6*y**2, 2*x],
        [6*y**2,    2*x, 2*y],
        [   2*x,    2*y,   0]])


def test_QR():
    A = Matrix([[1, 2], [2, 3]])
    Q, S = A.QRdecomposition()
    R = Rational
    assert Q == Matrix([
        [  5**R(-1, 2),  (R(2)/5)*(R(1)/5)**R(-1, 2)],
        [2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
    assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
    assert Q*S == A
    assert Q.T * Q == eye(2)

    A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q*R


def test_QR_non_square():
    A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q*R

    A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q*R


def test_nullspace():
    # first test reduced row-ech form
    R = Rational

    M = Matrix([[5, 7, 2,  1],
                [1, 6, 2, -1]])
    out, tmp = M.rref()
    assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
                          [0, 1,  R(8)/23, R(-6)/23]])

    M = Matrix([[-5, -1,  4, -3, -1],
                [ 1, -1, -1,  1,  0],
                [-1,  0,  0,  0,  0],
                [ 4,  1, -4,  3,  1],
                [-2,  0,  2, -2, -1]])
    assert M*M.nullspace()[0] == Matrix(5, 1, [0]*5)

    M = Matrix([[ 1,  3, 0,  2,  6, 3, 1],
                [-2, -6, 0, -2, -8, 3, 1],
                [ 3,  9, 0,  0,  6, 6, 2],
                [-1, -3, 0,  1,  0, 9, 3]])
    out, tmp = M.rref()
    assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
                          [0, 0, 0, 1, 2, 0, 0],
                          [0, 0, 0, 0, 0, 1, R(1)/3],
                          [0, 0, 0, 0, 0, 0, 0]])

    # now check the vectors
    basis = M.nullspace()
    assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
    assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
    assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
    assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])

    # issue 4797; just see that we can do it when rows > cols
    M = Matrix([[1, 2], [2, 4], [3, 6]])
    assert M.nullspace()


def test_columnspace():
    M = Matrix([[ 1,  2,  0,  2,  5],
                [-2, -5,  1, -1, -8],
                [ 0, -3,  3,  4,  1],
                [ 3,  6,  0, -7,  2]])

    # now check the vectors
    basis = M.columnspace()
    assert basis[0] == Matrix([1, -2, 0, 3])
    assert basis[1] == Matrix([2, -5, -3, 6])
    assert basis[2] == Matrix([2, -1, 4, -7])

    #check by columnspace definition
    a, b, c, d, e = symbols('a b c d e')
    X = Matrix([a, b, c, d, e])
    for i in range(len(basis)):
        eq=M*X-basis[i]
        assert len(solve(eq, X)) != 0

    #check if rank-nullity theorem holds
    assert M.rank() == len(basis)
    assert len(M.nullspace()) + len(M.columnspace()) == M.cols


def test_wronskian():
    assert wronskian([cos(x), sin(x)], x) == cos(x)**2 + sin(x)**2
    assert wronskian([exp(x), exp(2*x)], x) == exp(3*x)
    assert wronskian([exp(x), x], x) == exp(x) - x*exp(x)
    assert wronskian([1, x, x**2], x) == 2
    w1 = -6*exp(x)*sin(x)*x + 6*cos(x)*exp(x)*x**2 - 6*exp(x)*cos(x)*x - \
        exp(x)*cos(x)*x**3 + exp(x)*sin(x)*x**3
    assert wronskian([exp(x), cos(x), x**3], x).expand() == w1
    assert wronskian([exp(x), cos(x), x**3], x, method='berkowitz').expand() \
        == w1
    w2 = -x**3*cos(x)**2 - x**3*sin(x)**2 - 6*x*cos(x)**2 - 6*x*sin(x)**2
    assert wronskian([sin(x), cos(x), x**3], x).expand() == w2
    assert wronskian([sin(x), cos(x), x**3], x, method='berkowitz').expand() \
        == w2
    assert wronskian([], x) == 1


def test_eigen():
    R = Rational

    assert eye(3).charpoly(x) == Poly((x - 1)**3, x)
    assert eye(3).charpoly(y) == Poly((y - 1)**3, y)

    M = Matrix([[1, 0, 0],
                [0, 1, 0],
                [0, 0, 1]])

    assert M.eigenvals(multiple=False) == {S.One: 3}
    assert M.eigenvals(multiple=True) == [1, 1, 1]

    assert M.eigenvects() == (
        [(1, 3, [Matrix([1, 0, 0]),
                 Matrix([0, 1, 0]),
                 Matrix([0, 0, 1])])])

    assert M.left_eigenvects() == (
        [(1, 3, [Matrix([[1, 0, 0]]),
                 Matrix([[0, 1, 0]]),
                 Matrix([[0, 0, 1]])])])

    M = Matrix([[0, 1, 1],
                [1, 0, 0],
                [1, 1, 1]])

    assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}

    assert M.eigenvects() == (
        [
            (-1, 1, [Matrix([-1, 1, 0])]),
            ( 0, 1, [Matrix([0, -1, 1])]),
            ( 2, 1, [Matrix([R(2, 3), R(1, 3), 1])])
        ])

    assert M.left_eigenvects() == (
        [
            (-1, 1, [Matrix([[-2, 1, 1]])]),
            (0, 1, [Matrix([[-1, -1, 1]])]),
            (2, 1, [Matrix([[1, 1, 1]])])
        ])

    a = Symbol('a')
    M = Matrix([[a, 0],
                [0, 1]])

    assert M.eigenvals() == {a: 1, S.One: 1}

    M = Matrix([[1, -1],
                [1,  3]])
    assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
    assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])

    M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    a = R(15, 2)
    b = 3*33**R(1, 2)
    c = R(13, 2)
    d = (R(33, 8) + 3*b/8)
    e = (R(33, 8) - 3*b/8)

    def NS(e, n):
        return str(N(e, n))
    r = [
        (a - b/2, 1, [Matrix([(12 + 24/(c - b/2))/((c - b/2)*e) + 3/(c - b/2),
                              (6 + 12/(c - b/2))/e, 1])]),
        (      0, 1, [Matrix([1, -2, 1])]),
        (a + b/2, 1, [Matrix([(12 + 24/(c + b/2))/((c + b/2)*d) + 3/(c + b/2),
                              (6 + 12/(c + b/2))/d, 1])]),
    ]
    r1 = [(NS(r[i][0], 2), NS(r[i][1], 2),
        [NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
    r = M.eigenvects()
    r2 = [(NS(r[i][0], 2), NS(r[i][1], 2),
        [NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
    assert sorted(r1) == sorted(r2)

    eps = Symbol('eps', real=True)

    M = Matrix([[abs(eps), I*eps    ],
                [-I*eps,   abs(eps) ]])

    assert M.eigenvects() == (
        [
            ( 0, 1, [Matrix([[-I*eps/abs(eps)], [1]])]),
            ( 2*abs(eps), 1, [ Matrix([[I*eps/abs(eps)], [1]]) ] ),
        ])

    assert M.left_eigenvects() == (
        [
            (0, 1, [Matrix([[I*eps/Abs(eps), 1]])]),
            (2*Abs(eps), 1, [Matrix([[-I*eps/Abs(eps), 1]])])
        ])

    M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
    M._eigenvects = M.eigenvects(simplify=False)
    assert max(i.q for i in M._eigenvects[0][2][0]) > 1
    M._eigenvects = M.eigenvects(simplify=True)
    assert max(i.q for i in M._eigenvects[0][2][0]) == 1
    M = Matrix([[S(1)/4, 1], [1, 1]])
    assert M.eigenvects(simplify=True) == [
        (S(5)/8 + sqrt(73)/8, 1, [Matrix([[-S(3)/8 + sqrt(73)/8], [1]])]),
        (-sqrt(73)/8 + S(5)/8, 1, [Matrix([[-sqrt(73)/8 - S(3)/8], [1]])])]
    assert M.eigenvects(simplify=False) ==[(S(5)/8 + sqrt(73)/8, 1, [Matrix([
       [-1/(-sqrt(73)/8 - S(3)/8)],
       [                     1]])]), (-sqrt(73)/8 + S(5)/8, 1, [Matrix([
       [-1/(-S(3)/8 + sqrt(73)/8)],
       [                     1]])])]

    m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
    evals = {-sqrt(385)/20 + S(5)/4: 1, sqrt(385)/20 + S(5)/4: 1, S.Zero: 1}
    assert m.eigenvals() == evals
    nevals = list(sorted(m.eigenvals(rational=False).keys()))
    sevals = list(sorted(evals.keys()))
    assert all(abs(nevals[i] - sevals[i]) < 1e-9 for i in range(len(nevals)))

    # issue 10719
    assert Matrix([]).eigenvals() == {}
    assert Matrix([]).eigenvects() == []

    # issue 15119
    raises(NonSquareMatrixError, lambda : Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
    raises(NonSquareMatrixError, lambda : Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
    raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
    raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
    raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(error_when_incomplete = False))
    raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(error_when_incomplete = False))

    # issue 15125
    from sympy.core.function import count_ops
    q = Symbol("q", positive = True)
    m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
    assert count_ops(m.eigenvals(simplify=False)) > count_ops(m.eigenvals(simplify=True))
    assert count_ops(m.eigenvals(simplify=lambda x: x)) > count_ops(m.eigenvals(simplify=True))

    assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
    assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
    assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
    assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)

def test_subs():
    assert Matrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
    assert Matrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
        Matrix([[-1, 2], [-3, 4]])
    assert Matrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
        Matrix([[-1, 2], [-3, 4]])
    assert Matrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
        Matrix([[-1, 2], [-3, 4]])
    assert Matrix([x*y]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
        Matrix([(x - 1)*(y - 1)])

    for cls in classes:
        assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).subs(1, 2)

def test_xreplace():
    assert Matrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
        Matrix([[1, 5], [5, 4]])
    assert Matrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
        Matrix([[-1, 2], [-3, 4]])
    for cls in classes:
        assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).xreplace({1: 2})

def test_simplify():
    n = Symbol('n')
    f = Function('f')

    M = Matrix([[            1/x + 1/y,                 (x + x*y) / x  ],
                [ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
    M.simplify()
    assert M == Matrix([[ (x + y)/(x * y),                        1 + y ],
                        [           1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
    eq = (1 + x)**2
    M = Matrix([[eq]])
    M.simplify()
    assert M == Matrix([[eq]])
    M.simplify(ratio=oo) == M
    assert M == Matrix([[eq.simplify(ratio=oo)]])


def test_transpose():
    M = Matrix([[1, 2, 3, 4, 5, 6, 7, 8, 9, 0],
                [1, 2, 3, 4, 5, 6, 7, 8, 9, 0]])
    assert M.T == Matrix( [ [1, 1],
                            [2, 2],
                            [3, 3],
                            [4, 4],
                            [5, 5],
                            [6, 6],
                            [7, 7],
                            [8, 8],
                            [9, 9],
                            [0, 0] ])
    assert M.T.T == M
    assert M.T == M.transpose()


def test_conjugate():
    M = Matrix([[0, I, 5],
                [1, 2, 0]])

    assert M.T == Matrix([[0, 1],
                          [I, 2],
                          [5, 0]])

    assert M.C == Matrix([[0, -I, 5],
                          [1,  2, 0]])
    assert M.C == M.conjugate()

    assert M.H == M.T.C
    assert M.H == Matrix([[ 0, 1],
                          [-I, 2],
                          [ 5, 0]])


def test_conj_dirac():
    raises(AttributeError, lambda: eye(3).D)

    M = Matrix([[1, I, I, I],
                [0, 1, I, I],
                [0, 0, 1, I],
                [0, 0, 0, 1]])

    assert M.D == Matrix([[ 1,  0,  0,  0],
                          [-I,  1,  0,  0],
                          [-I, -I, -1,  0],
                          [-I, -I,  I, -1]])


def test_trace():
    M = Matrix([[1, 0, 0],
                [0, 5, 0],
                [0, 0, 8]])
    assert M.trace() == 14


def test_shape():
    M = Matrix([[x, 0, 0],
                [0, y, 0]])
    assert M.shape == (2, 3)


def test_col_row_op():
    M = Matrix([[x, 0, 0],
                [0, y, 0]])
    M.row_op(1, lambda r, j: r + j + 1)
    assert M == Matrix([[x,     0, 0],
                        [1, y + 2, 3]])

    M.col_op(0, lambda c, j: c + y**j)
    assert M == Matrix([[x + 1,     0, 0],
                        [1 + y, y + 2, 3]])

    # neither row nor slice give copies that allow the original matrix to
    # be changed
    assert M.row(0) == Matrix([[x + 1, 0, 0]])
    r1 = M.row(0)
    r1[0] = 42
    assert M[0, 0] == x + 1
    r1 = M[0, :-1]  # also testing negative slice
    r1[0] = 42
    assert M[0, 0] == x + 1
    c1 = M.col(0)
    assert c1 == Matrix([x + 1, 1 + y])
    c1[0] = 0
    assert M[0, 0] == x + 1
    c1 = M[:, 0]
    c1[0] = 42
    assert M[0, 0] == x + 1


def test_zip_row_op():
    for cls in classes[:2]: # XXX: immutable matrices don't support row ops
        M = cls.eye(3)
        M.zip_row_op(1, 0, lambda v, u: v + 2*u)
        assert M == cls([[1, 0, 0],
                         [2, 1, 0],
                         [0, 0, 1]])

        M = cls.eye(3)*2
        M[0, 1] = -1
        M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
        assert M == cls([[2, -1, 0],
                         [4,  0, 0],
                         [0,  0, 2]])

def test_issue_3950():
    m = Matrix([1, 2, 3])
    a = Matrix([1, 2, 3])
    b = Matrix([2, 2, 3])
    assert not (m in [])
    assert not (m in [1])
    assert m != 1
    assert m == a
    assert m != b


def test_issue_3981():
    class Index1(object):
        def __index__(self):
            return 1

    class Index2(object):
        def __index__(self):
            return 2
    index1 = Index1()
    index2 = Index2()

    m = Matrix([1, 2, 3])

    assert m[index2] == 3

    m[index2] = 5
    assert m[2] == 5

    m = Matrix([[1, 2, 3], [4, 5, 6]])
    assert m[index1, index2] == 6
    assert m[1, index2] == 6
    assert m[index1, 2] == 6

    m[index1, index2] = 4
    assert m[1, 2] == 4
    m[1, index2] = 6
    assert m[1, 2] == 6
    m[index1, 2] = 8
    assert m[1, 2] == 8


def test_evalf():
    a = Matrix([sqrt(5), 6])
    assert all(a.evalf()[i] == a[i].evalf() for i in range(2))
    assert all(a.evalf(2)[i] == a[i].evalf(2) for i in range(2))
    assert all(a.n(2)[i] == a[i].n(2) for i in range(2))


def test_is_symbolic():
    a = Matrix([[x, x], [x, x]])
    assert a.is_symbolic() is True
    a = Matrix([[1, 2, 3, 4], [5, 6, 7, 8]])
    assert a.is_symbolic() is False
    a = Matrix([[1, 2, 3, 4], [5, 6, x, 8]])
    assert a.is_symbolic() is True
    a = Matrix([[1, x, 3]])
    assert a.is_symbolic() is True
    a = Matrix([[1, 2, 3]])
    assert a.is_symbolic() is False
    a = Matrix([[1], [x], [3]])
    assert a.is_symbolic() is True
    a = Matrix([[1], [2], [3]])
    assert a.is_symbolic() is False


def test_is_upper():
    a = Matrix([[1, 2, 3]])
    assert a.is_upper is True
    a = Matrix([[1], [2], [3]])
    assert a.is_upper is False
    a = zeros(4, 2)
    assert a.is_upper is True


def test_is_lower():
    a = Matrix([[1, 2, 3]])
    assert a.is_lower is False
    a = Matrix([[1], [2], [3]])
    assert a.is_lower is True


def test_is_nilpotent():
    a = Matrix(4, 4, [0, 2, 1, 6, 0, 0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 0])
    assert a.is_nilpotent()
    a = Matrix([[1, 0], [0, 1]])
    assert not a.is_nilpotent()
    a = Matrix([])
    assert a.is_nilpotent()


def test_zeros_ones_fill():
    n, m = 3, 5

    a = zeros(n, m)
    a.fill( 5 )

    b = 5 * ones(n, m)

    assert a == b
    assert a.rows == b.rows == 3
    assert a.cols == b.cols == 5
    assert a.shape == b.shape == (3, 5)
    assert zeros(2) == zeros(2, 2)
    assert ones(2) == ones(2, 2)
    assert zeros(2, 3) == Matrix(2, 3, [0]*6)
    assert ones(2, 3) == Matrix(2, 3, [1]*6)


def test_empty_zeros():
    a = zeros(0)
    assert a == Matrix()
    a = zeros(0, 2)
    assert a.rows == 0
    assert a.cols == 2
    a = zeros(2, 0)
    assert a.rows == 2
    assert a.cols == 0


def test_issue_3749():
    a = Matrix([[x**2, x*y], [x*sin(y), x*cos(y)]])
    assert a.diff(x) == Matrix([[2*x, y], [sin(y), cos(y)]])
    assert Matrix([
        [x, -x, x**2],
        [exp(x), 1/x - exp(-x), x + 1/x]]).limit(x, oo) == \
        Matrix([[oo, -oo, oo], [oo, 0, oo]])
    assert Matrix([
        [(exp(x) - 1)/x, 2*x + y*x, x**x ],
        [1/x, abs(x), abs(sin(x + 1))]]).limit(x, 0) == \
        Matrix([[1, 0, 1], [oo, 0, sin(1)]])
    assert a.integrate(x) == Matrix([
        [Rational(1, 3)*x**3, y*x**2/2],
        [x**2*sin(y)/2, x**2*cos(y)/2]])


def test_inv_iszerofunc():
    A = eye(4)
    A.col_swap(0, 1)
    for method in "GE", "LU":
        assert A.inv(method=method, iszerofunc=lambda x: x == 0) == \
            A.inv(method="ADJ")


def test_jacobian_metrics():
    rho, phi = symbols("rho,phi")
    X = Matrix([rho*cos(phi), rho*sin(phi)])
    Y = Matrix([rho, phi])
    J = X.jacobian(Y)
    assert J == X.jacobian(Y.T)
    assert J == (X.T).jacobian(Y)
    assert J == (X.T).jacobian(Y.T)
    g = J.T*eye(J.shape[0])*J
    g = g.applyfunc(trigsimp)
    assert g == Matrix([[1, 0], [0, rho**2]])


def test_jacobian2():
    rho, phi = symbols("rho,phi")
    X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
    Y = Matrix([rho, phi])
    J = Matrix([
        [cos(phi), -rho*sin(phi)],
        [sin(phi),  rho*cos(phi)],
        [   2*rho,             0],
    ])
    assert X.jacobian(Y) == J


def test_issue_4564():
    X = Matrix([exp(x + y + z), exp(x + y + z), exp(x + y + z)])
    Y = Matrix([x, y, z])
    for i in range(1, 3):
        for j in range(1, 3):
            X_slice = X[:i, :]
            Y_slice = Y[:j, :]
            J = X_slice.jacobian(Y_slice)
            assert J.rows == i
            assert J.cols == j
            for k in range(j):
                assert J[:, k] == X_slice


def test_nonvectorJacobian():
    X = Matrix([[exp(x + y + z), exp(x + y + z)],
                [exp(x + y + z), exp(x + y + z)]])
    raises(TypeError, lambda: X.jacobian(Matrix([x, y, z])))
    X = X[0, :]
    Y = Matrix([[x, y], [x, z]])
    raises(TypeError, lambda: X.jacobian(Y))
    raises(TypeError, lambda: X.jacobian(Matrix([ [x, y], [x, z] ])))


def test_vec():
    m = Matrix([[1, 3], [2, 4]])
    m_vec = m.vec()
    assert m_vec.cols == 1
    for i in range(4):
        assert m_vec[i] == i + 1


def test_vech():
    m = Matrix([[1, 2], [2, 3]])
    m_vech = m.vech()
    assert m_vech.cols == 1
    for i in range(3):
        assert m_vech[i] == i + 1
    m_vech = m.vech(diagonal=False)
    assert m_vech[0] == 2

    m = Matrix([[1, x*(x + y)], [y*x + x**2, 1]])
    m_vech = m.vech(diagonal=False)
    assert m_vech[0] == x*(x + y)

    m = Matrix([[1, x*(x + y)], [y*x, 1]])
    m_vech = m.vech(diagonal=False, check_symmetry=False)
    assert m_vech[0] == y*x


def test_vech_errors():
    m = Matrix([[1, 3]])
    raises(ShapeError, lambda: m.vech())
    m = Matrix([[1, 3], [2, 4]])
    raises(ValueError, lambda: m.vech())
    raises(ShapeError, lambda: Matrix([ [1, 3] ]).vech())
    raises(ValueError, lambda: Matrix([ [1, 3], [2, 4] ]).vech())


def test_diag():
    a = Matrix([[1, 2], [2, 3]])
    b = Matrix([[3, x], [y, 3]])
    c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
    assert diag(a, b, b) == Matrix([
        [1, 2, 0, 0, 0, 0],
        [2, 3, 0, 0, 0, 0],
        [0, 0, 3, x, 0, 0],
        [0, 0, y, 3, 0, 0],
        [0, 0, 0, 0, 3, x],
        [0, 0, 0, 0, y, 3],
    ])
    assert diag(a, b, c) == Matrix([
        [1, 2, 0, 0, 0, 0, 0],
        [2, 3, 0, 0, 0, 0, 0],
        [0, 0, 3, x, 0, 0, 0],
        [0, 0, y, 3, 0, 0, 0],
        [0, 0, 0, 0, 3, x, 3],
        [0, 0, 0, 0, y, 3, z],
        [0, 0, 0, 0, x, y, z],
    ])
    assert diag(a, c, b) == Matrix([
        [1, 2, 0, 0, 0, 0, 0],
        [2, 3, 0, 0, 0, 0, 0],
        [0, 0, 3, x, 3, 0, 0],
        [0, 0, y, 3, z, 0, 0],
        [0, 0, x, y, z, 0, 0],
        [0, 0, 0, 0, 0, 3, x],
        [0, 0, 0, 0, 0, y, 3],
    ])
    a = Matrix([x, y, z])
    b = Matrix([[1, 2], [3, 4]])
    c = Matrix([[5, 6]])
    assert diag(a, 7, b, c) == Matrix([
        [x, 0, 0, 0, 0, 0],
        [y, 0, 0, 0, 0, 0],
        [z, 0, 0, 0, 0, 0],
        [0, 7, 0, 0, 0, 0],
        [0, 0, 1, 2, 0, 0],
        [0, 0, 3, 4, 0, 0],
        [0, 0, 0, 0, 5, 6],
    ])
    assert diag(1, [2, 3], [[4, 5]]) == Matrix([
        [1, 0, 0, 0],
        [0, 2, 0, 0],
        [0, 3, 0, 0],
        [0, 0, 4, 5]])


def test_get_diag_blocks1():
    a = Matrix([[1, 2], [2, 3]])
    b = Matrix([[3, x], [y, 3]])
    c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
    assert a.get_diag_blocks() == [a]
    assert b.get_diag_blocks() == [b]
    assert c.get_diag_blocks() == [c]


def test_get_diag_blocks2():
    a = Matrix([[1, 2], [2, 3]])
    b = Matrix([[3, x], [y, 3]])
    c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
    assert diag(a, b, b).get_diag_blocks() == [a, b, b]
    assert diag(a, b, c).get_diag_blocks() == [a, b, c]
    assert diag(a, c, b).get_diag_blocks() == [a, c, b]
    assert diag(c, c, b).get_diag_blocks() == [c, c, b]


def test_inv_block():
    a = Matrix([[1, 2], [2, 3]])
    b = Matrix([[3, x], [y, 3]])
    c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
    A = diag(a, b, b)
    assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), b.inv())
    A = diag(a, b, c)
    assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), c.inv())
    A = diag(a, c, b)
    assert A.inv(try_block_diag=True) == diag(a.inv(), c.inv(), b.inv())
    A = diag(a, a, b, a, c, a)
    assert A.inv(try_block_diag=True) == diag(
        a.inv(), a.inv(), b.inv(), a.inv(), c.inv(), a.inv())
    assert A.inv(try_block_diag=True, method="ADJ") == diag(
        a.inv(method="ADJ"), a.inv(method="ADJ"), b.inv(method="ADJ"),
        a.inv(method="ADJ"), c.inv(method="ADJ"), a.inv(method="ADJ"))


def test_creation_args():
    """
    Check that matrix dimensions can be specified using any reasonable type
    (see issue 4614).
    """
    raises(ValueError, lambda: zeros(3, -1))
    raises(TypeError, lambda: zeros(1, 2, 3, 4))
    assert zeros(long(3)) == zeros(3)
    assert zeros(Integer(3)) == zeros(3)
    assert zeros(3.) == zeros(3)
    assert eye(long(3)) == eye(3)
    assert eye(Integer(3)) == eye(3)
    assert eye(3.) == eye(3)
    assert ones(long(3), Integer(4)) == ones(3, 4)
    raises(TypeError, lambda: Matrix(5))
    raises(TypeError, lambda: Matrix(1, 2))


def test_diagonal_symmetrical():
    m = Matrix(2, 2, [0, 1, 1, 0])
    assert not m.is_diagonal()
    assert m.is_symmetric()
    assert m.is_symmetric(simplify=False)

    m = Matrix(2, 2, [1, 0, 0, 1])
    assert m.is_diagonal()

    m = diag(1, 2, 3)
    assert m.is_diagonal()
    assert m.is_symmetric()

    m = Matrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
    assert m == diag(1, 2, 3)

    m = Matrix(2, 3, zeros(2, 3))
    assert not m.is_symmetric()
    assert m.is_diagonal()

    m = Matrix(((5, 0), (0, 6), (0, 0)))
    assert m.is_diagonal()

    m = Matrix(((5, 0, 0), (0, 6, 0)))
    assert m.is_diagonal()

    m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
    assert m.is_symmetric()
    assert not m.is_symmetric(simplify=False)
    assert m.expand().is_symmetric(simplify=False)


def test_diagonalization():
    m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
    assert not m.is_diagonalizable()
    assert not m.is_symmetric()
    raises(NonSquareMatrixError, lambda: m.diagonalize())

    # diagonalizable
    m = diag(1, 2, 3)
    (P, D) = m.diagonalize()
    assert P == eye(3)
    assert D == m

    m = Matrix(2, 2, [0, 1, 1, 0])
    assert m.is_symmetric()
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D

    m = Matrix(2, 2, [1, 0, 0, 3])
    assert m.is_symmetric()
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D
    assert P == eye(2)
    assert D == m

    m = Matrix(2, 2, [1, 1, 0, 0])
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D

    m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D
    for i in P:
        assert i.as_numer_denom()[1] == 1

    m = Matrix(2, 2, [1, 0, 0, 0])
    assert m.is_diagonal()
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D
    assert P == Matrix([[0, 1], [1, 0]])

    # diagonalizable, complex only
    m = Matrix(2, 2, [0, 1, -1, 0])
    assert not m.is_diagonalizable(True)
    raises(MatrixError, lambda: m.diagonalize(True))
    assert m.is_diagonalizable()
    (P, D) = m.diagonalize()
    assert P.inv() * m * P == D

    # not diagonalizable
    m = Matrix(2, 2, [0, 1, 0, 0])
    assert not m.is_diagonalizable()
    raises(MatrixError, lambda: m.diagonalize())

    m = Matrix(3, 3, [-3, 1, -3, 20, 3, 10, 2, -2, 4])
    assert not m.is_diagonalizable()
    raises(MatrixError, lambda: m.diagonalize())

    # symbolic
    a, b, c, d = symbols('a b c d')
    m = Matrix(2, 2, [a, c, c, b])
    assert m.is_symmetric()
    assert m.is_diagonalizable()


@XFAIL
def test_eigen_vects():
    m = Matrix(2, 2, [1, 0, 0, I])
    raises(NotImplementedError, lambda: m.is_diagonalizable(True))
    # !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
    # see issue 5292
    assert not m.is_diagonalizable(True)
    raises(MatrixError, lambda: m.diagonalize(True))
    (P, D) = m.diagonalize(True)


def test_jordan_form():

    m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
    raises(NonSquareMatrixError, lambda: m.jordan_form())

    # diagonalizable
    m = Matrix(3, 3, [7, -12, 6, 10, -19, 10, 12, -24, 13])
    Jmust = Matrix(3, 3, [-1, 0, 0, 0, 1, 0, 0, 0, 1])
    P, J = m.jordan_form()
    assert Jmust == J
    assert Jmust == m.diagonalize()[1]

    # m = Matrix(3, 3, [0, 6, 3, 1, 3, 1, -2, 2, 1])
    # m.jordan_form()  # very long
    # m.jordan_form()  #

    # diagonalizable, complex only

    # Jordan cells
    # complexity: one of eigenvalues is zero
    m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
    # The blocks are ordered according to the value of their eigenvalues,
    # in order to make the matrix compatible with .diagonalize()
    Jmust = Matrix(3, 3, [2, 1, 0, 0, 2, 0, 0, 0, 2])
    P, J = m.jordan_form()
    assert Jmust == J

    # complexity: all of eigenvalues are equal
    m = Matrix(3, 3, [2, 6, -15, 1, 1, -5, 1, 2, -6])
    # Jmust = Matrix(3, 3, [-1, 0, 0, 0, -1, 1, 0, 0, -1])
    # same here see 1456ff
    Jmust = Matrix(3, 3, [-1, 1, 0, 0, -1, 0, 0, 0, -1])
    P, J = m.jordan_form()
    assert Jmust == J

    # complexity: two of eigenvalues are zero
    m = Matrix(3, 3, [4, -5, 2, 5, -7, 3, 6, -9, 4])
    Jmust = Matrix(3, 3, [0, 1, 0, 0, 0, 0, 0, 0, 1])
    P, J = m.jordan_form()
    assert Jmust == J

    m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5])
    Jmust = Matrix(4, 4, [2, 1, 0, 0,
                          0, 2, 0, 0,
              0, 0, 2, 1,
              0, 0, 0, 2]
              )
    P, J = m.jordan_form()
    assert Jmust == J

    m = Matrix(4, 4, [6, 2, -8, -6, -3, 2, 9, 6, 2, -2, -8, -6, -1, 0, 3, 4])
    # Jmust = Matrix(4, 4, [2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, -2])
    # same here see 1456ff
    Jmust = Matrix(4, 4, [-2, 0, 0, 0,
                           0, 2, 1, 0,
                           0, 0, 2, 0,
                           0, 0, 0, 2])
    P, J = m.jordan_form()
    assert Jmust == J

    m = Matrix(4, 4, [5, 4, 2, 1, 0, 1, -1, -1, -1, -1, 3, 0, 1, 1, -1, 2])
    assert not m.is_diagonalizable()
    Jmust = Matrix(4, 4, [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4])
    P, J = m.jordan_form()
    assert Jmust == J

    # checking for maximum precision to remain unchanged
    m = Matrix([[Float('1.0', precision=110), Float('2.0', precision=110)],
                [Float('3.14159265358979323846264338327', precision=110), Float('4.0', precision=110)]])
    P, J = m.jordan_form()
    for term in J._mat:
        if isinstance(term, Float):
            assert term._prec == 110


def test_jordan_form_complex_issue_9274():
    A = Matrix([[ 2,  4,  1,  0],
                [-4,  2,  0,  1],
                [ 0,  0,  2,  4],
                [ 0,  0, -4,  2]])
    p = 2 - 4*I;
    q = 2 + 4*I;
    Jmust1 = Matrix([[p, 1, 0, 0],
                     [0, p, 0, 0],
                     [0, 0, q, 1],
                     [0, 0, 0, q]])
    Jmust2 = Matrix([[q, 1, 0, 0],
                     [0, q, 0, 0],
                     [0, 0, p, 1],
                     [0, 0, 0, p]])
    P, J = A.jordan_form()
    assert J == Jmust1 or J == Jmust2
    assert simplify(P*J*P.inv()) == A

def test_issue_10220():
    # two non-orthogonal Jordan blocks with eigenvalue 1
    M = Matrix([[1, 0, 0, 1],
                [0, 1, 1, 0],
                [0, 0, 1, 1],
                [0, 0, 0, 1]])
    P, J = M.jordan_form()
    assert P == Matrix([[0, 1, 0, 1],
                        [1, 0, 0, 0],
                        [0, 1, 0, 0],
                        [0, 0, 1, 0]])
    assert J == Matrix([
                        [1, 1, 0, 0],
                        [0, 1, 1, 0],
                        [0, 0, 1, 0],
                        [0, 0, 0, 1]])


def test_Matrix_berkowitz_charpoly():
    UA, K_i, K_w = symbols('UA K_i K_w')

    A = Matrix([[-K_i - UA + K_i**2/(K_i + K_w),       K_i*K_w/(K_i + K_w)],
                [           K_i*K_w/(K_i + K_w), -K_w + K_w**2/(K_i + K_w)]])

    charpoly = A.charpoly(x)

    assert charpoly == \
        Poly(x**2 + (K_i*UA + K_w*UA + 2*K_i*K_w)/(K_i + K_w)*x +
        K_i*K_w*UA/(K_i + K_w), x, domain='ZZ(K_i,K_w,UA)')

    assert type(charpoly) is PurePoly

    A = Matrix([[1, 3], [2, 0]])
    assert A.charpoly() == A.charpoly(x) == PurePoly(x**2 - x - 6)

    A = Matrix([[1, 2], [x, 0]])
    p = A.charpoly(x)
    assert p.gen != x
    assert p.as_expr().subs(p.gen, x) == x**2 - 3*x


def test_exp():
    m = Matrix([[3, 4], [0, -2]])
    m_exp = Matrix([[exp(3), -4*exp(-2)/5 + 4*exp(3)/5], [0, exp(-2)]])
    assert m.exp() == m_exp
    assert exp(m) == m_exp

    m = Matrix([[1, 0], [0, 1]])
    assert m.exp() == Matrix([[E, 0], [0, E]])
    assert exp(m) == Matrix([[E, 0], [0, E]])

    m = Matrix([[1, -1], [1, 1]])
    assert m.exp() == Matrix([[E*cos(1), -E*sin(1)], [E*sin(1), E*cos(1)]])


def test_has():
    A = Matrix(((x, y), (2, 3)))
    assert A.has(x)
    assert not A.has(z)
    assert A.has(Symbol)

    A = A.subs(x, 2)
    assert not A.has(x)

def test_LUdecomposition_Simple_iszerofunc():
    # Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
    # matrices.LUdecomposition_Simple()
    magic_string = "I got passed in!"
    def goofyiszero(value):
        raise ValueError(magic_string)

    try:
        lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
    except ValueError as err:
        assert magic_string == err.args[0]
        return

    assert False

def test_LUdecomposition_iszerofunc():
    # Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
    # matrices.LUdecomposition_Simple()
    magic_string = "I got passed in!"
    def goofyiszero(value):
        raise ValueError(magic_string)

    try:
        l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
    except ValueError as err:
        assert magic_string == err.args[0]
        return

    assert False

def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero1():
    # Test if matrices._find_reasonable_pivot_naive()
    # finds a guaranteed non-zero pivot when the
    # some of the candidate pivots are symbolic expressions.
    # Keyword argument: simpfunc=None indicates that no simplifications
    # should be performed during the search.
    x = Symbol('x')
    column = Matrix(3, 1, [x, cos(x)**2 + sin(x)**2, Rational(1, 2)])
    pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
        _find_reasonable_pivot_naive(column)
    assert pivot_val == Rational(1, 2)

def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero2():
    # Test if matrices._find_reasonable_pivot_naive()
    # finds a guaranteed non-zero pivot when the
    # some of the candidate pivots are symbolic expressions.
    # Keyword argument: simpfunc=_simplify indicates that the search
    # should attempt to simplify candidate pivots.
    x = Symbol('x')
    column = Matrix(3, 1,
                    [x,
                     cos(x)**2+sin(x)**2+x**2,
                     cos(x)**2+sin(x)**2])
    pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
        _find_reasonable_pivot_naive(column, simpfunc=_simplify)
    assert pivot_val == 1

def test_find_reasonable_pivot_naive_simplifies():
    # Test if matrices._find_reasonable_pivot_naive()
    # simplifies candidate pivots, and reports
    # their offsets correctly.
    x = Symbol('x')
    column = Matrix(3, 1,
                    [x,
                     cos(x)**2+sin(x)**2+x,
                     cos(x)**2+sin(x)**2])
    pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
        _find_reasonable_pivot_naive(column, simpfunc=_simplify)

    assert len(simplified) == 2
    assert simplified[0][0] == 1
    assert simplified[0][1] == 1+x
    assert simplified[1][0] == 2
    assert simplified[1][1] == 1

def test_errors():
    raises(ValueError, lambda: Matrix([[1, 2], [1]]))
    raises(IndexError, lambda: Matrix([[1, 2]])[1.2, 5])
    raises(IndexError, lambda: Matrix([[1, 2]])[1, 5.2])
    raises(ValueError, lambda: randMatrix(3, c=4, symmetric=True))
    raises(ValueError, lambda: Matrix([1, 2]).reshape(4, 6))
    raises(ShapeError,
        lambda: Matrix([[1, 2], [3, 4]]).copyin_matrix([1, 0], Matrix([1, 2])))
    raises(TypeError, lambda: Matrix([[1, 2], [3, 4]]).copyin_list([0,
           1], set([])))
    raises(NonSquareMatrixError, lambda: Matrix([[1, 2, 3], [2, 3, 0]]).inv())
    raises(ShapeError,
        lambda: Matrix(1, 2, [1, 2]).row_join(Matrix([[1, 2], [3, 4]])))
    raises(
        ShapeError, lambda: Matrix([1, 2]).col_join(Matrix([[1, 2], [3, 4]])))
    raises(ShapeError, lambda: Matrix([1]).row_insert(1, Matrix([[1,
           2], [3, 4]])))
    raises(ShapeError, lambda: Matrix([1]).col_insert(1, Matrix([[1,
           2], [3, 4]])))
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).trace())
    raises(TypeError, lambda: Matrix([1]).applyfunc(1))
    raises(ShapeError, lambda: Matrix([1]).LUsolve(Matrix([[1, 2], [3, 4]])))
    raises(MatrixError, lambda: Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]
           ]).QRdecomposition())
    raises(MatrixError, lambda: Matrix(1, 2, [1, 2]).QRdecomposition())
    raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor(4, 5))
    raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor_submatrix(4, 5))
    raises(TypeError, lambda: Matrix([1, 2, 3]).cross(1))
    raises(TypeError, lambda: Matrix([1, 2, 3]).dot(1))
    raises(ShapeError, lambda: Matrix([1, 2, 3]).dot(Matrix([1, 2])))
    raises(ShapeError, lambda: Matrix([1, 2]).dot([]))
    raises(TypeError, lambda: Matrix([1, 2]).dot('a'))
    raises(SymPyDeprecationWarning, lambda: Matrix([[1, 2], [3, 4]]).dot(Matrix([[4, 3], [1, 2]])))
    raises(ShapeError, lambda: Matrix([1, 2]).dot([1, 2, 3]))
    raises(NonSquareMatrixError, lambda: Matrix([1, 2, 3]).exp())
    raises(ShapeError, lambda: Matrix([[1, 2], [3, 4]]).normalized())
    raises(ValueError, lambda: Matrix([1, 2]).inv(method='not a method'))
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_GE())
    raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_GE())
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_ADJ())
    raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_ADJ())
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_LU())
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).is_nilpotent())
    raises(NonSquareMatrixError, lambda: Matrix([1, 2]).det())
    raises(ValueError,
        lambda: Matrix([[1, 2], [3, 4]]).det(method='Not a real method'))
    raises(ValueError,
        lambda: hessian(Matrix([[1, 2], [3, 4]]), Matrix([[1, 2], [2, 1]])))
    raises(ValueError, lambda: hessian(Matrix([[1, 2], [3, 4]]), []))
    raises(ValueError, lambda: hessian(Symbol('x')**2, 'a'))
    raises(IndexError, lambda: eye(3)[5, 2])
    raises(IndexError, lambda: eye(3)[2, 5])
    M = Matrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
    raises(ValueError, lambda: M.det('method=LU_decomposition()'))
    V = Matrix([[10, 10, 10]])
    M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(ValueError, lambda: M.row_insert(4.7, V))
    M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(ValueError, lambda: M.col_insert(-4.2, V))

def test_len():
    assert len(Matrix()) == 0
    assert len(Matrix([[1, 2]])) == len(Matrix([[1], [2]])) == 2
    assert len(Matrix(0, 2, lambda i, j: 0)) == \
        len(Matrix(2, 0, lambda i, j: 0)) == 0
    assert len(Matrix([[0, 1, 2], [3, 4, 5]])) == 6
    assert Matrix([1]) == Matrix([[1]])
    assert not Matrix()
    assert Matrix() == Matrix([])


def test_integrate():
    A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2)))
    assert A.integrate(x) == \
        Matrix(((x, 4*x, x**2/2), (x*y, 2*x, 4*x), (10*x, 5*x, x**3/3)))
    assert A.integrate(y) == \
        Matrix(((y, 4*y, x*y), (y**2/2, 2*y, 4*y), (10*y, 5*y, y*x**2)))


def test_limit():
    A = Matrix(((1, 4, sin(x)/x), (y, 2, 4), (10, 5, x**2 + 1)))
    assert A.limit(x, 0) == Matrix(((1, 4, 1), (y, 2, 4), (10, 5, 1)))


def test_diff():
    A = MutableDenseMatrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
    assert A.diff(x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
    assert A.diff(y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))

    assert diff(A, x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
    assert diff(A, y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))

    A_imm = A.as_immutable()
    assert A_imm.diff(x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
    assert A_imm.diff(y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))

    assert diff(A_imm, x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
    assert diff(A_imm, y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))

    # Derive matrix by matrix:

    A = MutableDenseMatrix([[x, y], [z, t]])
    assert A.diff(A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
    assert diff(A, A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])

    A_imm = A.as_immutable()
    assert A_imm.diff(A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
    assert diff(A_imm, A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])

    # Derive a constant matrix:
    assert A.diff(a) == MutableDenseMatrix([[0, 0], [0, 0]])

    B = ImmutableDenseMatrix([a, b])
    assert A.diff(B) == Array(
        [[[
            [0,0],
            [0,0]
        ]],
        [[
            [0,0],
            [0,0]
        ]]])

    # Test diff with tuples:

    dB = B.diff([[a, b]])
    assert dB.shape == (2, 2, 1)
    assert dB == Array([[[1], [0]], [[0], [1]]])

    f = Function("f")
    fxyz = f(x, y, z)
    assert fxyz.diff([[x, y, z]]) == Array([fxyz.diff(x), fxyz.diff(y), fxyz.diff(z)])
    assert fxyz.diff(([x, y, z], 2)) == Array([
        [fxyz.diff(x, 2), fxyz.diff(x, y), fxyz.diff(x, z)],
        [fxyz.diff(x, y), fxyz.diff(y, 2), fxyz.diff(y, z)],
        [fxyz.diff(x, z), fxyz.diff(z, y), fxyz.diff(z, 2)],
    ])

    expr = sin(x)*exp(y)
    assert expr.diff([[x, y]]) == Array([cos(x)*exp(y), sin(x)*exp(y)])
    assert expr.diff(y, ((x, y),)) == Array([cos(x)*exp(y), sin(x)*exp(y)])
    assert expr.diff(x, ((x, y),)) == Array([-sin(x)*exp(y), cos(x)*exp(y)])
    assert expr.diff(((y, x),), [[x, y]]) == Array([[cos(x)*exp(y), -sin(x)*exp(y)], [sin(x)*exp(y), cos(x)*exp(y)]])

    # Test different notations:

    fxyz.diff(x).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[0, 1, 0]
    fxyz.diff(z).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[2, 1, 0]
    fxyz.diff([[x, y, z]], ((z, y, x),)) == Array([[fxyz.diff(i).diff(j) for i in (x, y, z)] for j in (z, y, x)])

    # Test scalar derived by matrix remains matrix:
    res = x.diff(Matrix([[x, y]]))
    assert isinstance(res, ImmutableDenseMatrix)
    assert res == Matrix([[1, 0]])
    res = (x**3).diff(Matrix([[x, y]]))
    assert isinstance(res, ImmutableDenseMatrix)
    assert res == Matrix([[3*x**2, 0]])


def test_getattr():
    A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
    raises(AttributeError, lambda: A.nonexistantattribute)
    assert getattr(A, 'diff')(x) == Matrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))


def test_hessenberg():
    A = Matrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
    assert A.is_upper_hessenberg
    A = A.T
    assert A.is_lower_hessenberg
    A[0, -1] = 1
    assert A.is_lower_hessenberg is False

    A = Matrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
    assert not A.is_upper_hessenberg

    A = zeros(5, 2)
    assert A.is_upper_hessenberg


def test_cholesky():
    raises(NonSquareMatrixError, lambda: Matrix((1, 2)).cholesky())
    raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).cholesky())
    raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).cholesky())
    raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).cholesky())
    A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
    assert A.cholesky() * A.cholesky().T == A
    assert A.cholesky().is_lower
    assert A.cholesky() == Matrix([[5, 0, 0], [3, 3, 0], [-1, 1, 3]])
    A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
    assert A.cholesky() == Matrix(((2, 0, 0), (I, 1, 0), (1 - I, 0, 3)))


def test_LDLdecomposition():
    raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
    raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
    raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
    raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
    A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
    L, D = A.LDLdecomposition()
    assert L * D * L.T == A
    assert L.is_lower
    assert L == Matrix([[1, 0, 0], [ S(3)/5, 1, 0], [S(-1)/5, S(1)/3, 1]])
    assert D.is_diagonal()
    assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
    A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
    L, D = A.LDLdecomposition()
    assert expand_mul(L * D * L.H) == A
    assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S(1)/2 - I/2, 0, 1)))
    assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))


def test_cholesky_solve():
    A = Matrix([[2, 3, 5],
                [3, 6, 2],
                [8, 3, 6]])
    x = Matrix(3, 1, [3, 7, 5])
    b = A*x
    soln = A.cholesky_solve(b)
    assert soln == x
    A = Matrix([[0, -1, 2],
                [5, 10, 7],
                [8,  3, 4]])
    x = Matrix(3, 1, [-1, 2, 5])
    b = A*x
    soln = A.cholesky_solve(b)
    assert soln == x
    A = Matrix(((9, 3*I), (-3*I, 5)))
    x = Matrix((-2, 1))
    b = A*x
    soln = A.cholesky_solve(b)
    assert expand_mul(soln) == x
    A = Matrix(((9*I, 3), (-3 + I, 5)))
    x = Matrix((2 + 3*I, -1))
    b = A*x
    soln = A.cholesky_solve(b)
    assert expand_mul(soln) == x


def test_LDLsolve():
    A = Matrix([[2, 3, 5],
                [3, 6, 2],
                [8, 3, 6]])
    x = Matrix(3, 1, [3, 7, 5])
    b = A*x
    soln = A.LDLsolve(b)
    assert soln == x
    A = Matrix([[0, -1, 2],
                [5, 10, 7],
                [8,  3, 4]])
    x = Matrix(3, 1, [-1, 2, 5])
    b = A*x
    soln = A.LDLsolve(b)
    assert soln == x
    A = Matrix(((9, 3*I), (-3*I, 5)))
    x = Matrix((-2, 1))
    b = A*x
    soln = A.LDLsolve(b)
    assert expand_mul(soln) == x
    A = Matrix(((9*I, 3), (-3 + I, 5)))
    x = Matrix((2 + 3*I, -1))
    b = A*x
    soln = A.cholesky_solve(b)
    assert expand_mul(soln) == x


def test_lower_triangular_solve():

    raises(NonSquareMatrixError,
        lambda: Matrix([1, 0]).lower_triangular_solve(Matrix([0, 1])))
    raises(ShapeError,
        lambda: Matrix([[1, 0], [0, 1]]).lower_triangular_solve(Matrix([1])))
    raises(ValueError,
        lambda: Matrix([[2, 1], [1, 2]]).lower_triangular_solve(
            Matrix([[1, 0], [0, 1]])))

    A = Matrix([[1, 0], [0, 1]])
    B = Matrix([[x, y], [y, x]])
    C = Matrix([[4, 8], [2, 9]])

    assert A.lower_triangular_solve(B) == B
    assert A.lower_triangular_solve(C) == C


def test_upper_triangular_solve():

    raises(NonSquareMatrixError,
        lambda: Matrix([1, 0]).upper_triangular_solve(Matrix([0, 1])))
    raises(TypeError,
        lambda: Matrix([[1, 0], [0, 1]]).upper_triangular_solve(Matrix([1])))
    raises(TypeError,
        lambda: Matrix([[2, 1], [1, 2]]).upper_triangular_solve(
            Matrix([[1, 0], [0, 1]])))

    A = Matrix([[1, 0], [0, 1]])
    B = Matrix([[x, y], [y, x]])
    C = Matrix([[2, 4], [3, 8]])

    assert A.upper_triangular_solve(B) == B
    assert A.upper_triangular_solve(C) == C


def test_diagonal_solve():
    raises(TypeError, lambda: Matrix([1, 1]).diagonal_solve(Matrix([1])))
    A = Matrix([[1, 0], [0, 1]])*2
    B = Matrix([[x, y], [y, x]])
    assert A.diagonal_solve(B) == B/2


def test_matrix_norm():
    # Vector Tests
    # Test columns and symbols
    x = Symbol('x', real=True)
    v = Matrix([cos(x), sin(x)])
    assert trigsimp(v.norm(2)) == 1
    assert v.norm(10) == Pow(cos(x)**10 + sin(x)**10, S(1)/10)

    # Test Rows
    A = Matrix([[5, Rational(3, 2)]])
    assert A.norm() == Pow(25 + Rational(9, 4), S(1)/2)
    assert A.norm(oo) == max(A._mat)
    assert A.norm(-oo) == min(A._mat)

    # Matrix Tests
    # Intuitive test
    A = Matrix([[1, 1], [1, 1]])
    assert A.norm(2) == 2
    assert A.norm(-2) == 0
    assert A.norm('frobenius') == 2
    assert eye(10).norm(2) == eye(10).norm(-2) == 1
    assert A.norm(oo) == 2

    # Test with Symbols and more complex entries
    A = Matrix([[3, y, y], [x, S(1)/2, -pi]])
    assert (A.norm('fro')
           == sqrt(S(37)/4 + 2*abs(y)**2 + pi**2 + x**2))

    # Check non-square
    A = Matrix([[1, 2, -3], [4, 5, Rational(13, 2)]])
    assert A.norm(2) == sqrt(S(389)/8 + sqrt(78665)/8)
    assert A.norm(-2) == S(0)
    assert A.norm('frobenius') == sqrt(389)/2

    # Test properties of matrix norms
    # http://en.wikipedia.org/wiki/Matrix_norm#Definition
    # Two matrices
    A = Matrix([[1, 2], [3, 4]])
    B = Matrix([[5, 5], [-2, 2]])
    C = Matrix([[0, -I], [I, 0]])
    D = Matrix([[1, 0], [0, -1]])
    L = [A, B, C, D]
    alpha = Symbol('alpha', real=True)

    for order in ['fro', 2, -2]:
        # Zero Check
        assert zeros(3).norm(order) == S(0)
        # Check Triangle Inequality for all Pairs of Matrices
        for X in L:
            for Y in L:
                dif = (X.norm(order) + Y.norm(order) -
                    (X + Y).norm(order))
                assert (dif >= 0)
        # Scalar multiplication linearity
        for M in [A, B, C, D]:
            dif = simplify((alpha*M).norm(order) -
                    abs(alpha) * M.norm(order))
            assert dif == 0

    # Test Properties of Vector Norms
    # http://en.wikipedia.org/wiki/Vector_norm
    # Two column vectors
    a = Matrix([1, 1 - 1*I, -3])
    b = Matrix([S(1)/2, 1*I, 1])
    c = Matrix([-1, -1, -1])
    d = Matrix([3, 2, I])
    e = Matrix([Integer(1e2), Rational(1, 1e2), 1])
    L = [a, b, c, d, e]
    alpha = Symbol('alpha', real=True)

    for order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity, pi]:
        # Zero Check
        if order > 0:
            assert Matrix([0, 0, 0]).norm(order) == S(0)
        # Triangle inequality on all pairs
        if order >= 1:  # Triangle InEq holds only for these norms
            for X in L:
                for Y in L:
                    dif = (X.norm(order) + Y.norm(order) -
                        (X + Y).norm(order))
                    assert simplify(dif >= 0) is S.true
        # Linear to scalar multiplication
        if order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity]:
            for X in L:
                dif = simplify((alpha*X).norm(order) -
                    (abs(alpha) * X.norm(order)))
                assert dif == 0

    # ord=1
    M = Matrix(3, 3, [1, 3, 0, -2, -1, 0, 3, 9, 6])
    assert M.norm(1) == 13


def test_condition_number():
    x = Symbol('x', real=True)
    A = eye(3)
    A[0, 0] = 10
    A[2, 2] = S(1)/10
    assert A.condition_number() == 100

    A[1, 1] = x
    assert A.condition_number() == Max(10, Abs(x)) / Min(S(1)/10, Abs(x))

    M = Matrix([[cos(x), sin(x)], [-sin(x), cos(x)]])
    Mc = M.condition_number()
    assert all(Float(1.).epsilon_eq(Mc.subs(x, val).evalf()) for val in
        [Rational(1, 5), Rational(1, 2), Rational(1, 10), pi/2, pi, 7*pi/4 ])

    #issue 10782
    assert Matrix([]).condition_number() == 0


def test_equality():
    A = Matrix(((1, 2, 3), (4, 5, 6), (7, 8, 9)))
    B = Matrix(((9, 8, 7), (6, 5, 4), (3, 2, 1)))
    assert A == A[:, :]
    assert not A != A[:, :]
    assert not A == B
    assert A != B
    assert A != 10
    assert not A == 10

    # A SparseMatrix can be equal to a Matrix
    C = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
    D = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
    assert C == D
    assert not C != D


def test_col_join():
    assert eye(3).col_join(Matrix([[7, 7, 7]])) == \
        Matrix([[1, 0, 0],
                [0, 1, 0],
                [0, 0, 1],
                [7, 7, 7]])


def test_row_insert():
    r4 = Matrix([[4, 4, 4]])
    for i in range(-4, 5):
        l = [1, 0, 0]
        l.insert(i, 4)
        assert flatten(eye(3).row_insert(i, r4).col(0).tolist()) == l


def test_col_insert():
    c4 = Matrix([4, 4, 4])
    for i in range(-4, 5):
        l = [0, 0, 0]
        l.insert(i, 4)
        assert flatten(zeros(3).col_insert(i, c4).row(0).tolist()) == l


def test_normalized():
    assert Matrix([3, 4]).normalized() == \
        Matrix([Rational(3, 5), Rational(4, 5)])


def test_print_nonzero():
    assert capture(lambda: eye(3).print_nonzero()) == \
        '[X  ]\n[ X ]\n[  X]\n'
    assert capture(lambda: eye(3).print_nonzero('.')) == \
        '[.  ]\n[ . ]\n[  .]\n'


def test_zeros_eye():
    assert Matrix.eye(3) == eye(3)
    assert Matrix.zeros(3) == zeros(3)
    assert ones(3, 4) == Matrix(3, 4, [1]*12)

    i = Matrix([[1, 0], [0, 1]])
    z = Matrix([[0, 0], [0, 0]])
    for cls in classes:
        m = cls.eye(2)
        assert i == m  # but m == i will fail if m is immutable
        assert i == eye(2, cls=cls)
        assert type(m) == cls
        m = cls.zeros(2)
        assert z == m
        assert z == zeros(2, cls=cls)
        assert type(m) == cls


def test_is_zero():
    assert Matrix().is_zero
    assert Matrix([[0, 0], [0, 0]]).is_zero
    assert zeros(3, 4).is_zero
    assert not eye(3).is_zero
    assert Matrix([[x, 0], [0, 0]]).is_zero == None
    assert SparseMatrix([[x, 0], [0, 0]]).is_zero == None
    assert ImmutableMatrix([[x, 0], [0, 0]]).is_zero == None
    assert ImmutableSparseMatrix([[x, 0], [0, 0]]).is_zero == None
    assert Matrix([[x, 1], [0, 0]]).is_zero == False
    a = Symbol('a', nonzero=True)
    assert Matrix([[a, 0], [0, 0]]).is_zero == False


def test_rotation_matrices():
    # This tests the rotation matrices by rotating about an axis and back.
    theta = pi/3
    r3_plus = rot_axis3(theta)
    r3_minus = rot_axis3(-theta)
    r2_plus = rot_axis2(theta)
    r2_minus = rot_axis2(-theta)
    r1_plus = rot_axis1(theta)
    r1_minus = rot_axis1(-theta)
    assert r3_minus*r3_plus*eye(3) == eye(3)
    assert r2_minus*r2_plus*eye(3) == eye(3)
    assert r1_minus*r1_plus*eye(3) == eye(3)

    # Check the correctness of the trace of the rotation matrix
    assert r1_plus.trace() == 1 + 2*cos(theta)
    assert r2_plus.trace() == 1 + 2*cos(theta)
    assert r3_plus.trace() == 1 + 2*cos(theta)

    # Check that a rotation with zero angle doesn't change anything.
    assert rot_axis1(0) == eye(3)
    assert rot_axis2(0) == eye(3)
    assert rot_axis3(0) == eye(3)


def test_DeferredVector():
    assert str(DeferredVector("vector")[4]) == "vector[4]"
    assert sympify(DeferredVector("d")) == DeferredVector("d")

def test_DeferredVector_not_iterable():
    assert not iterable(DeferredVector('X'))

def test_DeferredVector_Matrix():
    raises(TypeError, lambda: Matrix(DeferredVector("V")))

def test_GramSchmidt():
    R = Rational
    m1 = Matrix(1, 2, [1, 2])
    m2 = Matrix(1, 2, [2, 3])
    assert GramSchmidt([m1, m2]) == \
        [Matrix(1, 2, [1, 2]), Matrix(1, 2, [R(2)/5, R(-1)/5])]
    assert GramSchmidt([m1.T, m2.T]) == \
        [Matrix(2, 1, [1, 2]), Matrix(2, 1, [R(2)/5, R(-1)/5])]
    # from wikipedia
    assert GramSchmidt([Matrix([3, 1]), Matrix([2, 2])], True) == [
        Matrix([3*sqrt(10)/10, sqrt(10)/10]),
        Matrix([-sqrt(10)/10, 3*sqrt(10)/10])]


def test_casoratian():
    assert casoratian([1, 2, 3, 4], 1) == 0
    assert casoratian([1, 2, 3, 4], 1, zero=False) == 0


def test_zero_dimension_multiply():
    assert (Matrix()*zeros(0, 3)).shape == (0, 3)
    assert zeros(3, 0)*zeros(0, 3) == zeros(3, 3)
    assert zeros(0, 3)*zeros(3, 0) == Matrix()


def test_slice_issue_2884():
    m = Matrix(2, 2, range(4))
    assert m[1, :] == Matrix([[2, 3]])
    assert m[-1, :] == Matrix([[2, 3]])
    assert m[:, 1] == Matrix([[1, 3]]).T
    assert m[:, -1] == Matrix([[1, 3]]).T
    raises(IndexError, lambda: m[2, :])
    raises(IndexError, lambda: m[2, 2])


def test_slice_issue_3401():
    assert zeros(0, 3)[:, -1].shape == (0, 1)
    assert zeros(3, 0)[0, :] == Matrix(1, 0, [])


def test_copyin():
    s = zeros(3, 3)
    s[3] = 1
    assert s[:, 0] == Matrix([0, 1, 0])
    assert s[3] == 1
    assert s[3: 4] == [1]
    s[1, 1] = 42
    assert s[1, 1] == 42
    assert s[1, 1:] == Matrix([[42, 0]])
    s[1, 1:] = Matrix([[5, 6]])
    assert s[1, :] == Matrix([[1, 5, 6]])
    s[1, 1:] = [[42, 43]]
    assert s[1, :] == Matrix([[1, 42, 43]])
    s[0, 0] = 17
    assert s[:, :1] == Matrix([17, 1, 0])
    s[0, 0] = [1, 1, 1]
    assert s[:, 0] == Matrix([1, 1, 1])
    s[0, 0] = Matrix([1, 1, 1])
    assert s[:, 0] == Matrix([1, 1, 1])
    s[0, 0] = SparseMatrix([1, 1, 1])
    assert s[:, 0] == Matrix([1, 1, 1])


def test_invertible_check():
    # sometimes a singular matrix will have a pivot vector shorter than
    # the number of rows in a matrix...
    assert Matrix([[1, 2], [1, 2]]).rref() == (Matrix([[1, 2], [0, 0]]), (0,))
    raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inv())
    m = Matrix([
        [-1, -1,  0],
        [ x,  1,  1],
        [ 1,  x, -1],
    ])
    assert len(m.rref()[1]) != m.rows
    # in addition, unless simplify=True in the call to rref, the identity
    # matrix will be returned even though m is not invertible
    assert m.rref()[0] != eye(3)
    assert m.rref(simplify=signsimp)[0] != eye(3)
    raises(ValueError, lambda: m.inv(method="ADJ"))
    raises(ValueError, lambda: m.inv(method="GE"))
    raises(ValueError, lambda: m.inv(method="LU"))


@XFAIL
def test_issue_3959():
    x, y = symbols('x, y')
    e = x*y
    assert e.subs(x, Matrix([3, 5, 3])) == Matrix([3, 5, 3])*y


def test_issue_5964():
    assert str(Matrix([[1, 2], [3, 4]])) == 'Matrix([[1, 2], [3, 4]])'


def test_issue_7604():
    x, y = symbols(u"x y")
    assert sstr(Matrix([[x, 2*y], [y**2, x + 3]])) == \
        'Matrix([\n[   x,   2*y],\n[y**2, x + 3]])'


def test_is_Identity():
    assert eye(3).is_Identity
    assert eye(3).as_immutable().is_Identity
    assert not zeros(3).is_Identity
    assert not ones(3).is_Identity
    # issue 6242
    assert not Matrix([[1, 0, 0]]).is_Identity
    # issue 8854
    assert SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1}).is_Identity
    assert not SparseMatrix(2,3, range(6)).is_Identity
    assert not SparseMatrix(3,3, {(0,0):1, (1,1):1}).is_Identity
    assert not SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1, (0,1):2, (0,2):3}).is_Identity


def test_dot():
    assert ones(1, 3).dot(ones(3, 1)) == 3
    assert ones(1, 3).dot([1, 1, 1]) == 3
    assert Matrix([1, 2, 3]).dot(Matrix([1, 2, 3])) == 14


def test_dual():
    B_x, B_y, B_z, E_x, E_y, E_z = symbols(
        'B_x B_y B_z E_x E_y E_z', real=True)
    F = Matrix((
        (   0,  E_x,  E_y,  E_z),
        (-E_x,    0,  B_z, -B_y),
        (-E_y, -B_z,    0,  B_x),
        (-E_z,  B_y, -B_x,    0)
    ))
    Fd = Matrix((
        (  0, -B_x, -B_y, -B_z),
        (B_x,    0,  E_z, -E_y),
        (B_y, -E_z,    0,  E_x),
        (B_z,  E_y, -E_x,    0)
    ))
    assert F.dual().equals(Fd)
    assert eye(3).dual().equals(zeros(3))
    assert F.dual().dual().equals(-F)


def test_anti_symmetric():
    assert Matrix([1, 2]).is_anti_symmetric() is False
    m = Matrix(3, 3, [0, x**2 + 2*x + 1, y, -(x + 1)**2, 0, x*y, -y, -x*y, 0])
    assert m.is_anti_symmetric() is True
    assert m.is_anti_symmetric(simplify=False) is False
    assert m.is_anti_symmetric(simplify=lambda x: x) is False

    # tweak to fail
    m[2, 1] = -m[2, 1]
    assert m.is_anti_symmetric() is False
    # untweak
    m[2, 1] = -m[2, 1]

    m = m.expand()
    assert m.is_anti_symmetric(simplify=False) is True
    m[0, 0] = 1
    assert m.is_anti_symmetric() is False


def test_normalize_sort_diogonalization():
    A = Matrix(((1, 2), (2, 1)))
    P, Q = A.diagonalize(normalize=True)
    assert P*P.T == P.T*P == eye(P.cols)
    P, Q = A.diagonalize(normalize=True, sort=True)
    assert P*P.T == P.T*P == eye(P.cols)
    assert P*Q*P.inv() == A


def test_issue_5321():
    raises(ValueError, lambda: Matrix([[1, 2, 3], Matrix(0, 1, [])]))


def test_issue_5320():
    assert Matrix.hstack(eye(2), 2*eye(2)) == Matrix([
        [1, 0, 2, 0],
        [0, 1, 0, 2]
    ])
    assert Matrix.vstack(eye(2), 2*eye(2)) == Matrix([
        [1, 0],
        [0, 1],
        [2, 0],
        [0, 2]
    ])
    cls = SparseMatrix
    assert cls.hstack(cls(eye(2)), cls(2*eye(2))) == Matrix([
        [1, 0, 2, 0],
        [0, 1, 0, 2]
    ])

def test_issue_11944():
    A = Matrix([[1]])
    AIm = sympify(A)
    assert Matrix.hstack(AIm, A) == Matrix([[1, 1]])
    assert Matrix.vstack(AIm, A) == Matrix([[1], [1]])

def test_cross():
    a = [1, 2, 3]
    b = [3, 4, 5]
    col = Matrix([-2, 4, -2])
    row = col.T

    def test(M, ans):
        assert ans == M
        assert type(M) == cls
    for cls in classes:
        A = cls(a)
        B = cls(b)
        test(A.cross(B), col)
        test(A.cross(B.T), col)
        test(A.T.cross(B.T), row)
        test(A.T.cross(B), row)
    raises(ShapeError, lambda:
        Matrix(1, 2, [1, 1]).cross(Matrix(1, 2, [1, 1])))


def test_hash():
    for cls in classes[-2:]:
        s = {cls.eye(1), cls.eye(1)}
        assert len(s) == 1 and s.pop() == cls.eye(1)
    # issue 3979
    for cls in classes[:2]:
        assert not isinstance(cls.eye(1), Hashable)


@XFAIL
def test_issue_3979():
    # when this passes, delete this and change the [1:2]
    # to [:2] in the test_hash above for issue 3979
    cls = classes[0]
    raises(AttributeError, lambda: hash(cls.eye(1)))


def test_adjoint():
    dat = [[0, I], [1, 0]]
    ans = Matrix([[0, 1], [-I, 0]])
    for cls in classes:
        assert ans == cls(dat).adjoint()

def test_simplify_immutable():
    from sympy import simplify, sin, cos
    assert simplify(ImmutableMatrix([[sin(x)**2 + cos(x)**2]])) == \
                    ImmutableMatrix([[1]])

def test_rank():
    from sympy.abc import x
    m = Matrix([[1, 2], [x, 1 - 1/x]])
    assert m.rank() == 2
    n = Matrix(3, 3, range(1, 10))
    assert n.rank() == 2
    p = zeros(3)
    assert p.rank() == 0

def test_issue_11434():
    ax, ay, bx, by, cx, cy, dx, dy, ex, ey, t0, t1 = \
        symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
    M = Matrix([[ax, ay, ax*t0, ay*t0, 0],
                [bx, by, bx*t0, by*t0, 0],
                [cx, cy, cx*t0, cy*t0, 1],
                [dx, dy, dx*t0, dy*t0, 1],
                [ex, ey, 2*ex*t1 - ex*t0, 2*ey*t1 - ey*t0, 0]])
    assert M.rank() == 4

def test_rank_regression_from_so():
    # see:
    # http://stackoverflow.com/questions/19072700/why-does-sympy-give-me-the-wrong-answer-when-i-row-reduce-a-symbolic-matrix

    nu, lamb = symbols('nu, lambda')
    A = Matrix([[-3*nu,         1,                  0,  0],
                [ 3*nu, -2*nu - 1,                  2,  0],
                [    0,      2*nu, (-1*nu) - lamb - 2,  3],
                [    0,         0,          nu + lamb, -3]])
    expected_reduced = Matrix([[1, 0, 0, 1/(nu**2*(-lamb - nu))],
                               [0, 1, 0,    3/(nu*(-lamb - nu))],
                               [0, 0, 1,         3/(-lamb - nu)],
                               [0, 0, 0,                      0]])
    expected_pivots = (0, 1, 2)

    reduced, pivots = A.rref()

    assert simplify(expected_reduced - reduced) == zeros(*A.shape)
    assert pivots == expected_pivots

def test_replace():
    from sympy import symbols, Function, Matrix
    F, G = symbols('F, G', cls=Function)
    K = Matrix(2, 2, lambda i, j: G(i+j))
    M = Matrix(2, 2, lambda i, j: F(i+j))
    N = M.replace(F, G)
    assert N == K

def test_replace_map():
    from sympy import symbols, Function, Matrix
    F, G = symbols('F, G', cls=Function)
    K = Matrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1)\
    : G(1)}), (G(2), {F(2): G(2)})])
    M = Matrix(2, 2, lambda i, j: F(i+j))
    N = M.replace(F, G, True)
    assert N == K

def test_atoms():
    m = Matrix([[1, 2], [x, 1 - 1/x]])
    assert m.atoms() == {S(1),S(2),S(-1), x}
    assert m.atoms(Symbol) == {x}

@slow
def test_pinv():
    # Pseudoinverse of an invertible matrix is the inverse.
    A1 = Matrix([[a, b], [c, d]])
    assert simplify(A1.pinv()) == simplify(A1.inv())
    # Test the four properties of the pseudoinverse for various matrices.
    As = [Matrix([[13, 104], [2212, 3], [-3, 5]]),
          Matrix([[1, 7, 9], [11, 17, 19]]),
          Matrix([a, b])]
    for A in As:
        A_pinv = A.pinv()
        AAp = A * A_pinv
        ApA = A_pinv * A
        assert simplify(AAp * A) == A
        assert simplify(ApA * A_pinv) == A_pinv
        assert AAp.H == AAp
        assert ApA.H == ApA

def test_pinv_solve():
    # Fully determined system (unique result, identical to other solvers).
    A = Matrix([[1, 5], [7, 9]])
    B = Matrix([12, 13])
    assert A.pinv_solve(B) == A.cholesky_solve(B)
    assert A.pinv_solve(B) == A.LDLsolve(B)
    assert A.pinv_solve(B) == Matrix([sympify('-43/26'), sympify('71/26')])
    assert A * A.pinv() * B == B
    # Fully determined, with two-dimensional B matrix.
    B = Matrix([[12, 13, 14], [15, 16, 17]])
    assert A.pinv_solve(B) == A.cholesky_solve(B)
    assert A.pinv_solve(B) == A.LDLsolve(B)
    assert A.pinv_solve(B) == Matrix([[-33, -37, -41], [69, 75, 81]]) / 26
    assert A * A.pinv() * B == B
    # Underdetermined system (infinite results).
    A = Matrix([[1, 0, 1], [0, 1, 1]])
    B = Matrix([5, 7])
    solution = A.pinv_solve(B)
    w = {}
    for s in solution.atoms(Symbol):
        # Extract dummy symbols used in the solution.
        w[s.name] = s
    assert solution == Matrix([[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 1],
                               [w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 3],
                               [-w['w0_0']/3 - w['w1_0']/3 + w['w2_0']/3 + 4]])
    assert A * A.pinv() * B == B
    # Overdetermined system (least squares results).
    A = Matrix([[1, 0], [0, 0], [0, 1]])
    B = Matrix([3, 2, 1])
    assert A.pinv_solve(B) == Matrix([3, 1])
    # Proof the solution is not exact.
    assert A * A.pinv() * B != B

@XFAIL
def test_pinv_rank_deficient():
    # Test the four properties of the pseudoinverse for various matrices.
    As = [Matrix([[1, 1, 1], [2, 2, 2]]),
          Matrix([[1, 0], [0, 0]])]
    for A in As:
        A_pinv = A.pinv()
        AAp = A * A_pinv
        ApA = A_pinv * A
        assert simplify(AAp * A) == A
        assert simplify(ApA * A_pinv) == A_pinv
        assert AAp.H == AAp
        assert ApA.H == ApA
    # Test solving with rank-deficient matrices.
    A = Matrix([[1, 0], [0, 0]])
    # Exact, non-unique solution.
    B = Matrix([3, 0])
    solution = A.pinv_solve(B)
    w1 = solution.atoms(Symbol).pop()
    assert w1.name == 'w1_0'
    assert solution == Matrix([3, w1])
    assert A * A.pinv() * B == B
    # Least squares, non-unique solution.
    B = Matrix([3, 1])
    solution = A.pinv_solve(B)
    w1 = solution.atoms(Symbol).pop()
    assert w1.name == 'w1_0'
    assert solution == Matrix([3, w1])
    assert A * A.pinv() * B != B


def test_gauss_jordan_solve():

    # Square, full rank, unique solution
    A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
    b = Matrix([3, 6, 9])
    sol, params = A.gauss_jordan_solve(b)
    assert sol == Matrix([[-1], [2], [0]])
    assert params == Matrix(0, 1, [])

    # Square, reduced rank, parametrized solution
    A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    b = Matrix([3, 6, 9])
    sol, params, freevar = A.gauss_jordan_solve(b, freevar=True)
    w = {}
    for s in sol.atoms(Symbol):
        # Extract dummy symbols used in the solution.
        w[s.name] = s
    assert sol == Matrix([[w['tau0'] - 1], [-2*w['tau0'] + 2], [w['tau0']]])
    assert params == Matrix([[w['tau0']]])
    assert freevar == [2]

    # Square, reduced rank, parametrized solution
    A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
    b = Matrix([0, 0, 0])
    sol, params = A.gauss_jordan_solve(b)
    w = {}
    for s in sol.atoms(Symbol):
        w[s.name] = s
    assert sol == Matrix([[-2*w['tau0'] - 3*w['tau1']],
                         [w['tau0']], [w['tau1']]])
    assert params == Matrix([[w['tau0']], [w['tau1']]])

    # Square, reduced rank, parametrized solution
    A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
    b = Matrix([0, 0, 0])
    sol, params = A.gauss_jordan_solve(b)
    w = {}
    for s in sol.atoms(Symbol):
        w[s.name] = s
    assert sol == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
    assert params == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])

    # Square, reduced rank, no solution
    A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
    b = Matrix([0, 0, 1])
    raises(ValueError, lambda: A.gauss_jordan_solve(b))

    # Rectangular, tall, full rank, unique solution
    A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
    b = Matrix([0, 0, 1, 0])
    sol, params = A.gauss_jordan_solve(b)
    assert sol == Matrix([[-S(1)/2], [0], [S(1)/6]])
    assert params == Matrix(0, 1, [])

    # Rectangular, tall, full rank, no solution
    A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
    b = Matrix([0, 0, 0, 1])
    raises(ValueError, lambda: A.gauss_jordan_solve(b))

    # Rectangular, tall, reduced rank, parametrized solution
    A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
    b = Matrix([0, 0, 0, 1])
    sol, params = A.gauss_jordan_solve(b)
    w = {}
    for s in sol.atoms(Symbol):
        w[s.name] = s
    assert sol == Matrix([[-3*w['tau0'] + 5], [-1], [w['tau0']]])
    assert params == Matrix([[w['tau0']]])

    # Rectangular, tall, reduced rank, no solution
    A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
    b = Matrix([0, 0, 1, 1])
    raises(ValueError, lambda: A.gauss_jordan_solve(b))

    # Rectangular, wide, full rank, parametrized solution
    A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 1, 12]])
    b = Matrix([1, 1, 1])
    sol, params = A.gauss_jordan_solve(b)
    w = {}
    for s in sol.atoms(Symbol):
        w[s.name] = s
    assert sol == Matrix([[2*w['tau0'] - 1], [-3*w['tau0'] + 1], [0],
                         [w['tau0']]])
    assert params == Matrix([[w['tau0']]])

    # Rectangular, wide, reduced rank, parametrized solution
    A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
    b = Matrix([0, 1, 0])
    sol, params = A.gauss_jordan_solve(b)
    w = {}
    for s in sol.atoms(Symbol):
        w[s.name] = s
    assert sol == Matrix([[w['tau0'] + 2*w['tau1'] + 1/S(2)],
                         [-2*w['tau0'] - 3*w['tau1'] - 1/S(4)],
                         [w['tau0']], [w['tau1']]])
    assert params == Matrix([[w['tau0']], [w['tau1']]])
    # watch out for clashing symbols
    x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
    M = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
    A = M[:, :-1]
    b = M[:, -1:]
    sol, params = A.gauss_jordan_solve(b)
    assert params == Matrix(3, 1, [x0, x1, x2])
    assert sol == Matrix(5, 1, [x1, 0, x0, _x0, x2])

    # Rectangular, wide, reduced rank, no solution
    A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
    b = Matrix([1, 1, 1])
    raises(ValueError, lambda: A.gauss_jordan_solve(b))


def test_issue_7201():
    assert ones(0, 1) + ones(0, 1) == Matrix(0, 1, [])
    assert ones(1, 0) + ones(1, 0) == Matrix(1, 0, [])

def test_free_symbols():
    for M in ImmutableMatrix, ImmutableSparseMatrix, Matrix, SparseMatrix:
        assert M([[x], [0]]).free_symbols == {x}

def test_from_ndarray():
    """See issue 7465."""
    try:
        from numpy import array
    except ImportError:
        skip('NumPy must be available to test creating matrices from ndarrays')

    assert Matrix(array([1, 2, 3])) == Matrix([1, 2, 3])
    assert Matrix(array([[1, 2, 3]])) == Matrix([[1, 2, 3]])
    assert Matrix(array([[1, 2, 3], [4, 5, 6]])) == \
        Matrix([[1, 2, 3], [4, 5, 6]])
    assert Matrix(array([x, y, z])) == Matrix([x, y, z])
    raises(NotImplementedError, lambda: Matrix(array([[
        [1, 2], [3, 4]], [[5, 6], [7, 8]]])))

def test_hermitian():
    a = Matrix([[1, I], [-I, 1]])
    assert a.is_hermitian
    a[0, 0] = 2*I
    assert a.is_hermitian is False
    a[0, 0] = x
    assert a.is_hermitian is None
    a[0, 1] = a[1, 0]*I
    assert a.is_hermitian is False

def test_doit():
    a = Matrix([[Add(x,x, evaluate=False)]])
    assert a[0] != 2*x
    assert a.doit() == Matrix([[2*x]])

def test_issue_9457_9467_9876():
    # for row_del(index)
    M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    M.row_del(1)
    assert M == Matrix([[1, 2, 3], [3, 4, 5]])
    N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    N.row_del(-2)
    assert N == Matrix([[1, 2, 3], [3, 4, 5]])
    O = Matrix([[1, 2, 3], [5, 6, 7], [9, 10, 11]])
    O.row_del(-1)
    assert O == Matrix([[1, 2, 3], [5, 6, 7]])
    P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(IndexError, lambda: P.row_del(10))
    Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(IndexError, lambda: Q.row_del(-10))

    # for col_del(index)
    M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    M.col_del(1)
    assert M == Matrix([[1, 3], [2, 4], [3, 5]])
    N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    N.col_del(-2)
    assert N == Matrix([[1, 3], [2, 4], [3, 5]])
    P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(IndexError, lambda: P.col_del(10))
    Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
    raises(IndexError, lambda: Q.col_del(-10))

def test_issue_9422():
    x, y = symbols('x y', commutative=False)
    a, b = symbols('a b')
    M = eye(2)
    M1 = Matrix(2, 2, [x, y, y, z])
    assert y*x*M != x*y*M
    assert b*a*M == a*b*M
    assert x*M1 != M1*x
    assert a*M1 == M1*a
    assert y*x*M == Matrix([[y*x, 0], [0, y*x]])


def test_issue_10770():
    M = Matrix([])
    a = ['col_insert', 'row_join'], Matrix([9, 6, 3])
    b = ['row_insert', 'col_join'], a[1].T
    c = ['row_insert', 'col_insert'], Matrix([[1, 2], [3, 4]])
    for ops, m in (a, b, c):
        for op in ops:
            f = getattr(M, op)
            new = f(m) if 'join' in op else f(42, m)
            assert new == m and id(new) != id(m)


def test_issue_10658():
    A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    assert A.extract([0, 1, 2], [True, True, False]) == \
        Matrix([[1, 2], [4, 5], [7, 8]])
    assert A.extract([0, 1, 2], [True, False, False]) == Matrix([[1], [4], [7]])
    assert A.extract([True, False, False], [0, 1, 2]) == Matrix([[1, 2, 3]])
    assert A.extract([True, False, True], [0, 1, 2]) == \
        Matrix([[1, 2, 3], [7, 8, 9]])
    assert A.extract([0, 1, 2], [False, False, False]) == Matrix(3, 0, [])
    assert A.extract([False, False, False], [0, 1, 2]) == Matrix(0, 3, [])
    assert A.extract([True, False, True], [False, True, False]) == \
        Matrix([[2], [8]])

def test_opportunistic_simplification():
    # this test relates to issue #10718, #9480, #11434

    # issue #9480
    m = Matrix([[-5 + 5*sqrt(2), -5], [-5*sqrt(2)/2 + 5, -5*sqrt(2)/2]])
    assert m.rank() == 1

    # issue #10781
    m = Matrix([[3+3*sqrt(3)*I, -9],[4,-3+3*sqrt(3)*I]])
    assert simplify(m.rref()[0] - Matrix([[1, -9/(3 + 3*sqrt(3)*I)], [0, 0]])) == zeros(2, 2)

    # issue #11434
    ax,ay,bx,by,cx,cy,dx,dy,ex,ey,t0,t1 = symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
    m = Matrix([[ax,ay,ax*t0,ay*t0,0],[bx,by,bx*t0,by*t0,0],[cx,cy,cx*t0,cy*t0,1],[dx,dy,dx*t0,dy*t0,1],[ex,ey,2*ex*t1-ex*t0,2*ey*t1-ey*t0,0]])
    assert m.rank() == 4

def test_partial_pivoting():
    # example from https://en.wikipedia.org/wiki/Pivot_element
    # partial pivoting with back subsitution gives a perfect result
    # naive pivoting give an error ~1e-13, so anything better than
    # 1e-15 is good
    mm=Matrix([[0.003 ,59.14, 59.17],[ 5.291, -6.13,46.78]])
    assert (mm.rref()[0] - Matrix([[1.0,   0, 10.0], [  0, 1.0,  1.0]])).norm() < 1e-15

    # issue #11549
    m_mixed = Matrix([[6e-17, 1.0, 4],[ -1.0,   0, 8],[    0,   0, 1]])
    m_float = Matrix([[6e-17, 1.0, 4.],[ -1.0,   0., 8.],[    0.,   0., 1.]])
    m_inv = Matrix([[  0,    -1.0,  8.0],[1.0, 6.0e-17, -4.0],[  0,       0,    1]])
    # this example is numerically unstable and involves a matrix with a norm >= 8,
    # this comparing the difference of the results with 1e-15 is numerically sound.
    assert (m_mixed.inv() - m_inv).norm() < 1e-15
    assert (m_float.inv() - m_inv).norm() < 1e-15

def test_iszero_substitution():
    """ When doing numerical computations, all elements that pass
    the iszerofunc test should be set to numerically zero if they
    aren't already. """

    # Matrix from issue #9060
    m = Matrix([[0.9, -0.1, -0.2, 0],[-0.8, 0.9, -0.4, 0],[-0.1, -0.8, 0.6, 0]])
    m_rref = m.rref(iszerofunc=lambda x: abs(x)<6e-15)[0]
    m_correct = Matrix([[1.0,   0, -0.301369863013699, 0],[  0, 1.0, -0.712328767123288, 0],[  0,   0,                  0, 0]])
    m_diff = m_rref - m_correct
    assert m_diff.norm() < 1e-15
    # if a zero-substitution wasn't made, this entry will be -1.11022302462516e-16
    assert m_rref[2,2] == 0


@slow
def test_issue_11238():
    from sympy import Point
    xx = 8*tan(13*pi/45)/(tan(13*pi/45) + sqrt(3))
    yy = (-8*sqrt(3)*tan(13*pi/45)**2 + 24*tan(13*pi/45))/(-3 + tan(13*pi/45)**2)
    p1 = Point(0, 0)
    p2 = Point(1, -sqrt(3))
    p0 = Point(xx,yy)
    m1 = Matrix([p1 - simplify(p0), p2 - simplify(p0)])
    m2 = Matrix([p1 - p0, p2 - p0])
    m3 = Matrix([simplify(p1 - p0), simplify(p2 - p0)])

    assert m1.rank(simplify=True) == 1
    assert m2.rank(simplify=True) == 1
    assert m3.rank(simplify=True) == 1

def test_as_real_imag():
    m1 = Matrix(2,2,[1,2,3,4])
    m2 = m1*S.ImaginaryUnit
    m3 = m1 + m2

    for kls in classes:
        a,b = kls(m3).as_real_imag()
        assert list(a) == list(m1)
        assert list(b) == list(m1)

def test_deprecated():
    # Maintain tests for deprecated functions.  We must capture
    # the deprecation warnings.  When the deprecated functionality is
    # removed, the corresponding tests should be removed.
    with warnings.catch_warnings():
        warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
        m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
        P, Jcells = m.jordan_cells()
        assert Jcells[1] == Matrix(1, 1, [2])
        assert Jcells[0] == Matrix(2, 2, [2, 1, 0, 2])
        assert Matrix([[1,2],[3,4]]).dot(Matrix([[1,3],[4,5]])) == [10, 19, 14, 28]


def test_issue_14489():
    from sympy import Mod
    A = Matrix([-1, 1, 2])
    B = Matrix([10, 20, -15])

    assert Mod(A, 3) == Matrix([2, 1, 2])
    assert Mod(B, 4) == Matrix([2, 0, 1])

def test_issue_14517():
    M = Matrix([
        [   0, 10*I,    10*I,       0],
        [10*I,    0,       0,    10*I],
        [10*I,    0, 5 + 2*I,    10*I],
        [   0, 10*I,    10*I, 5 + 2*I]])
    ev = M.eigenvals()
    # test one random eigenvalue, the computation is a little slow
    test_ev = random.choice(list(ev.keys()))
    assert (M - test_ev*eye(4)).det() == 0

def test_issue_14943():
    # Test that __array__ accepts the optional dtype argument
    try:
        from numpy import array
    except ImportError:
        skip('NumPy must be available to test creating matrices from ndarrays')

    M = Matrix([[1,2], [3,4]])
    assert array(M, dtype=float).dtype.name == 'float64'

def test_issue_8240():
    # Eigenvalues of large triangular matrices
    n = 200

    diagonal_variables = [Symbol('x%s' % i) for i in range(n)]
    M = [[0 for i in range(n)] for j in range(n)]
    for i in range(n):
        M[i][i] = diagonal_variables[i]
    M = Matrix(M)

    eigenvals = M.eigenvals()
    assert len(eigenvals) == n
    for i in range(n):
        assert eigenvals[diagonal_variables[i]] == 1

    eigenvals = M.eigenvals(multiple=True)
    assert set(eigenvals) == set(diagonal_variables)

    # with multiplicity
    M = Matrix([[x, 0, 0], [1, y, 0], [2, 3, x]])
    eigenvals = M.eigenvals()
    assert eigenvals == {x: 2, y: 1}

    eigenvals = M.eigenvals(multiple=True)
    assert len(eigenvals) == 3
    assert eigenvals.count(x) == 2
    assert eigenvals.count(y) == 1
