Compute matrix eigenpairs
Analyze a square system's modal behavior.
SciPy · Function
linalg.eig
Solves ordinary or generalized eigenvalue problems for square matrices. It returns eigenvalues and, optionally, left and normalized right eigenvectors.
How it works
eig solves the ordinary problem A v = w v or, when b is supplied, the generalized problem A v = w B v. It can return normalized right eigenvectors, optional left eigenvectors, or homogeneous eigenvalue coordinates, while flags control input checking and possible in-place workspace use.
Syntax
scipy.linalg.eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True, homogeneous_eigvals=False)Parameters
| Name | Type | Description |
|---|---|---|
a | (M, M) array_like | Square input matrix for the eigenvalue problem. |
b | (M, M) array_like | Optional right-hand-side square matrix for the generalized problem. |
left | bool, optional | Whether to compute left eigenvectors. |
right | bool, optional | Whether to compute right eigenvectors. |
overwrite_a | bool, optional | Allow overwriting a when possible to reduce workspace use. |
overwrite_b | bool, optional | Allow overwriting b when possible to reduce workspace use. |
check_finite | bool, optional | Check inputs for finite values before computation. |
homogeneous_eigvals | bool, optional | Return generalized eigenvalues in homogeneous coordinates. |
Returns
Depending on left and right, returns the eigenvalues w, optional left eigenvectors vl, and optional normalized right eigenvectors vr; w uses homogeneous coordinates when homogeneous_eigvals=True.
Key points
- Use
eigfor square-matrix eigenproblems, either ordinary or generalized. - The routine supports eigenvalues together with right eigenvectors, left eigenvectors, or both.
- Each returned right eigenvector is stored as a column of
vrand is normalized. - When requested, left eigenvectors are arranged by eigenvalue in columns of
vl. - Pass
bto define the second matrix in the generalized equation. - The eigenvalue output normally has one value per matrix dimension; homogeneous output uses a different shape.
- The left-eigenvector output is conditional on setting
left=True.
Common mistakes
- Left eigenvectors returned with
left=Trueare not normalized, unlike the right eigenvectors.
Example: Compute eigenvalues and right eigenvectors
import numpy as np
from scipy.linalg import eig
A = np.array([[1.0, 2.0], [2.0, 1.0]])
w, vr = eig(A)
print(w)
print(vr)Computes the eigenvalues and normalized right eigenvectors of a real square matrix.
When to use it
Handle generalized eigenproblems
Solve A v = w B v for two square matrices.
Compute left eigenvectors
Obtain dual vectors for non-Hermitian matrices.
Use homogeneous eigenvalues
Represent potentially infinite generalized eigenvalues safely.
Interview questions
QWhat problems does scipy.linalg.eig solve?
It solves an ordinary or generalized eigenvalue problem. With only a, it finds w and vectors satisfying A v = w v; with b, it solves A v = w B v.
Follow-up: How are left eigenvectors characterized? They satisfy A.H vl[:, i] = w[i].conj() B.H vl[:, i], and they are returned only when left=True.
QWhat does the default return value contain?
By default, the function returns eigenvalues and normalized right eigenvectors. The eigenvector associated with w[i] is the column vr[:, i].
Follow-up: What changes when right=False? The right-eigenvector array is omitted, so the return structure contains only the requested eigenvalue and left-eigenvector results.
QWhat are b and homogeneous_eigvals used for?
b is the right-hand-side matrix for a generalized problem and must have shape (M, M). homogeneous_eigvals=True instead returns eigenvalues in homogeneous coordinates, represented with an extra leading dimension.
Follow-up: Why use homogeneous coordinates? They allow generalized eigenvalues that may otherwise be infinite to be represented without converting them to ordinary finite scalars.
Related
scipy.linalg.eighscipy.linalg.eigvalsscipy.linalg.eigvalsh