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.
For concreteness, consider the matrix
[[2, x], [x, 3]]
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.
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()
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
.
See also this section of the SymPy tutorial.