Diagonalize parametric matrix

[matteoettam09]

Hello,

is it possible to get a parametric eigendecomposition of a matrix whose elements are a mix of known numerical values and parametric variables, expressed as ndpoly entries?

Thank you.

[jonathf]

Currently no. I am not even sure if I understand how that would work. In the governing equation A x = lambda x do we assume all terms are polynomial scalar/vectors/matrices?

If you know how to approach such a problem and know of reading matterial I can look at, I can let you know if it would be feasable to implement or not.

[matteoettam09]

I think both the eigenvalue matrix and the eigenvector matrix (and its inverse) are in general a function of polynomial scalars appearing inside the given matrix, so yes, if I understand your question correctly.

Unfortunately, I have no idea about how to deal with the problem, it's just something that came out of my research.

[jonathf]

I figured a way to solve this, but it requires a few new functions that are not currently implemented. Stay tuned and I'll get back to this thread when all needed compoents are in place.

[matteoettam09]

Great! Thank you.

[jonathf]

So here is working example that should work in version 1.1.3:

import numpy as np
import numpoly as nu

# q1, q2 are parameters, q0 reserved for calculations as lambda
q0, q1, q2 = nu.variable(3)

# parametric matrix to find eigen vectors/values from
mat = nu.polynomial([[1, q1], [q2, 1]])

# characteristic eq == 0
char_expr = nu.det(mat-np.eye(len(mat))*q0)

# evaluate parameters and extract eigen values
eig_vals = nu.roots(nu.polynomial(char_expr(q1=3, q2=1)))

Temporary caveates:

• q0 has to be reserved to the "lambda" variable.
• char_expr needs to be cast to nu.polynomial to ensure end polynomial is in face 1D before root algorithm is used.

Let me know if this is what you were looking for.

[matteoettam09]

Yes, thank you, this is exactly was I was looking for! I will start working on it.