add 共轭上三角的转化

%% Cell type:code id: tags:

``` 

from sympy import *

x = symbols ('x')

f = symbols('f', cls=Function)

dsolve(f(x).diff(x)- cos(f(x)), f(x))

```

%% Output

    [Eq(f(x), pi - asin(1/tanh(C1 + x))), Eq(f(x), asin(1/tanh(C1 + x)))]

%% Cell type:code id: tags:

``` 

# 计算所有满足 M^2=I 的整系数矩阵, 无法使用这一方法计算 M^3

from sympy import *

a, b, c, d = symbols('a b c d');

M=Matrix([[a,b],[c,d]]);

K=M*M

N=Matrix([[1,0],[0,1]]);

solve(Eq(K,N),(a,b,c,d))

```

%% Output

    [(-1, 0, 0, -1),

     (-1, 0, 0, 1),

     (-1, 0, c, 1),

     (1, 0, 0, -1),

     (1, 0, 0, 1),

     (1, 0, c, -1),

     (-d, (1 - d**2)/c, c, d)]

%% Cell type:code id: tags:

``` 

# 计算所有满足 M^3=I 的整系数矩阵

from sympy import *

a, b, c, d = symbols('a b c d');

M=Matrix([[a,b],[c,d]]);

K=M**3

N=Matrix([[1,0],[0,1]]);

solve(Eq(K,N),(a,b,c,d))

```

%% Cell type:code id: tags:

``` 

# 计算共轭上三角，对应线性函数变化

from sympy import *

a, b, c, d = symbols('a b c d');

M = Matrix([[a,b],[(1-a**2)/b,-a]])

S = Matrix([[-b/(1+a),1],[1,0]])

T=S.inv()

T*M*S

```

%% Output

$\displaystyle \left[\begin{matrix}- a - \frac{1 - a^{2}}{a + 1} & \frac{1 - a^{2}}{b}\\- \frac{a b}{a + 1} + b - \frac{b \left(a + \frac{1 - a^{2}}{a + 1}\right)}{a + 1} & a + \frac{1 - a^{2}}{a + 1}\end{matrix}\right]$

    Matrix([

    [                              -a - (1 - a**2)/(a + 1),           (1 - a**2)/b],

    [-a*b/(a + 1) + b - b*(a + (1 - a**2)/(a + 1))/(a + 1), a + (1 - a**2)/(a + 1)]])

%% Cell type:code id: tags:

``` 

# 共轭上三角实例

from sympy import *

M=Matrix([[3,-2],[4,-3]])

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

K=N.inv()

K*M*N

```

%% Output

$\displaystyle \left[\begin{matrix}1 & 4\\0 & -1\end{matrix}\right]$

    Matrix([

    [1,  4],

    [0, -1]])

%% Cell type:code id: tags:

``` 

# 验证 m=-1/(1+x) 虽然满足 m^3=1 但对应矩阵表示不满足.

from sympy import *

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

M**3

```

%% Output

$\displaystyle \left[\begin{matrix}-1 & 0\\0 & -1\end{matrix}\right]$

    Matrix([

    [-1,  0],

    [ 0, -1]])

%% Cell type:code id: tags:

``` 

```