NumPy · Function

np.linalg.eig

Computes the eigenvalues and right eigenvectors of a square array. It returns paired eigenvalues and normalized eigenvectors, including support for batched square matrices.

How it works

Pass a square array to obtain a namedtuple containing eigenvalues and corresponding right eigenvectors. Each eigenvector is returned as a normalized column, so column i pairs with eigenvalue i; stacked leading dimensions are supported.

Syntax

numpy.linalg.eig(a)

Parameters

Name	Type	Description
`a`	`array`	Matrix or batch of matrices with shape (..., M, M) for which eigenvalues and right eigenvectors are computed.

Returns

A namedtuple with eigenvalues of shape (..., M) and eigenvectors of shape (..., M, M). Eigenvalues are not necessarily ordered; eigenvectors are normalized and column i corresponds to eigenvalues[i].

Key points

Use np.linalg.eig to compute both eigenvalues and right eigenvectors for a square matrix.
The argument is the matrix or batch of matrices to decompose.
The eigenvalues output includes repeated values when an eigenvalue has multiplicity greater than one.
Eigenvalues use a complex dtype when needed, but may be cast to a real dtype when their imaginary parts are zero.
Returned eigenvectors are normalized to unit length.
In the two-dimensional case, eigenvectors are columns, and column i corresponds to eigenvalues[i].
The general square-array eigensolver is backed by the _geev LAPACK routines.

Common mistakes

Do not assume the returned eigenvalues are sorted; preserve the eigenvalue–eigenvector pairing by using the same index.
The operation can raise LinAlgError when the eigenvalue calculation fails to converge.
For real input, complex eigenvalues appear in conjugate pairs, while real results may be represented with zero imaginary parts.

Example: Compute eigenpairs

import numpy as np

A = np.array([[1, 2], [2, 1]])
w, v = np.linalg.eig(A)
print(w)
print(v)
print(np.allclose(A @ v, v * w))

Computes eigenvalues and verifies each returned eigenvector equation.

When to use it

Matrix spectral analysis

Analyze linear transformations or matrix dynamics by examining characteristic values and directions.

Batched eigendecomposition

Process multiple square matrices at once while retaining leading batch dimensions.

Eigenpair validation

Verify numerical solutions by checking the eigenvector equation for returned pairs.

Interview questions

QWhat does np.linalg.eig return?

It computes the eigenvalues and right eigenvectors of a square array. The result is a namedtuple with eigenvalues and eigenvectors fields.

Follow-up: How do you match an eigenvalue to its vector? Use the same index: eigenvectors[:, i] corresponds to eigenvalues[i].

QWhat input shapes are supported?

The input has shape (..., M, M), allowing optional leading batch dimensions. The final two dimensions must form square matrices.

Follow-up: What shape does a batched result have? Eigenvalues retain the leading dimensions and have shape (..., M); eigenvectors have shape (..., M, M).

QHow are the returned eigenvectors organized?

Each returned eigenvector is normalized to unit length, and the vectors are stored as columns. The eigenvalue and vector at each index form a corresponding eigenpair.

Follow-up: How can an eigenpair be checked numerically? Verify A @ v[:, i] is close to eigenvalues[i] * v[:, i].

numpy.linalg.eigvals
numpy.linalg.eigh
numpy.linalg.eigvalsh
numpy.linalg.svd

Source: numpy.linalg.eig. Checked 2026-10-15.