Eigenvalues
Notation:
solve
given by numpy.linalg.eighfor diagonal matrix and numpy.linalg.eig for non.
Singular value
Notation:
Eigenvector
where is Eigenvector and is Eigenvalues.
Applications
Problem transformations
- Shift, where is scalar.
- Inversion, if nonsingular and .
- Powers, .
- Polynomial, where is polynomial.
Example
For is symmetric matrix, can be done by
where is Eigenvector matrix
ev, v = np.linalg.eigh(A)
expmA = v @ np.diag(np.exp(ev)) @ v^T