对角化符号矩阵 [英] Diagonalize symbolic matrix

问题描述

我需要用python对角化符号矩阵.在Mathematica中可以轻松完成，但是使用模块numpy.linalg时会遇到问题.

I need to diagonalize a symbolic matrix with python. In Mathematica it can be done easily, but when using the module numpy.linalg I get problems.

为具体起见，请考虑矩阵

For concreteness, consider the matrix

[[2, x], [x, 3]]

其中，x是符号变量.我想我会遇到问题，因为numpy包是为数值计算而不是符号提供的，但是我找不到如何用sympy做到这一点.

where x is a symbolic variable. I guess I get problems because the numpy package is provided for numerical computations, not symbolic, but I cannot find how to do it with sympy.

推荐答案

您可以根据特征值进行计算，但是实际上有一种方法可以为您做到这一点，

You can compute it from the eigenvalues, but there is actually a method that will do it for you, diagonalize

In [13]: M.diagonalize()
Out[13]:
⎛                                        ⎡     __________                       ⎤⎞
⎜                                        ⎢    ╱    2                            ⎥⎟
⎜⎡      -2⋅x                2⋅x       ⎤  ⎢  ╲╱  4⋅x  + 1    5                   ⎥⎟
⎜⎢─────────────────  ─────────────────⎥, ⎢- ───────────── + ─          0        ⎥⎟
⎜⎢   __________         __________    ⎥  ⎢        2         2                   ⎥⎟
⎜⎢  ╱    2             ╱    2         ⎥  ⎢                                      ⎥⎟
⎜⎢╲╱  4⋅x  + 1  - 1  ╲╱  4⋅x  + 1  + 1⎥  ⎢                        __________    ⎥⎟
⎜⎢                                    ⎥  ⎢                       ╱    2         ⎥⎟
⎜⎣        1                  1        ⎦  ⎢                     ╲╱  4⋅x  + 1    5⎥⎟
⎜                                        ⎢         0           ───────────── + ─⎥⎟
⎝                                        ⎣                           2         2⎦⎠

M.diagonalize()返回一对矩阵(P, D)，使得M = P*D*P**-1.如果由于矩阵不可对角化或由于solve()找不到特征多项式的所有根而不能计算出足够多的特征值，则会提高MatrixError.

M.diagonalize() returns a pair of matrices (P, D) such that M = P*D*P**-1. If it can't compute enough eigenvalues, either because the matrix is not diagonalizable or because solve() can't find all the roots of the characteristic polynomial, it will raise MatrixError.

另请参见本节 SymPy教程.

See also this section of the SymPy tutorial.

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