Diagonalize the symbolic matrix

Question

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

For concreteness, we consider the matrix

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

where x is a symbolic variable. I suppose I'm having problems because the numpy package is provided for numerical computation, not symbolic, but I cannot find how to do this with sympy.

Answer 1

You can calculate it from eigenvalues, but actually there is a method that will do this 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() returns a pair of matrices (P, D) such that M = P*D*P**-1 . If he cannot calculate sufficient eigenvalues, either because the matrix is ​​not diagonalizable, or because solve() cannot find all the roots of the characteristic polynomial, it will raise a MatrixError .

See also this section of the SymPy tutorial.

Answer 2

Assuming that the matrix is ​​diagonalizable, you can get the eigenvectors and eigenvalues ​​through

 from sympy import * x = Symbol('x') M = Matrix([[2,x],[x,3]]) print M.eigenvects() print M.eigenvals() 

Donation:

 [(-sqrt(4*x**2 + 1)/2 + 5/2, 1, [[-x/(sqrt(4*x**2 + 1)/2 - 1/2)] [ 1]]), (sqrt(4*x**2 + 1)/2 + 5/2, 1, [[-x/(-sqrt(4*x**2 + 1)/2 - 1/2)] [ 1]])] {sqrt(4*x**2 + 1)/2 + 5/2: 1, -sqrt(4*x**2 + 1)/2 + 5/2: 1} 

You should check the documentation , there are many other expansions.

Note that not every matrix is ​​diagonalizable, but you can put each matrix in the Jordan Normal Form using the sympy .jordan_form .