.. _routines.linalg:

.. module:: numpy.linalg

Linear algebra

The NumPy linear algebra functions rely on BLAS and LAPACK to provide efficient low level implementations of standard linear algebra algorithms. Those libraries may be provided by NumPy itself using C versions of a subset of their reference implementations but, when possible, highly optimized libraries that take advantage of specialized processor functionality are preferred. Examples of such libraries are OpenBLAS_, MKL (TM), and ATLAS. Because those libraries are multithreaded and processor dependent, environmental variables and external packages such as threadpoolctl_ may be needed to control the number of threads or specify the processor architecture.

.. _OpenBLAS: https://www.openblas.net/ .. _threadpoolctl: https://github.com/joblib/threadpoolctl

The SciPy library also contains a ~scipy.linalg submodule, and there is overlap in the functionality provided by the SciPy and NumPy submodules. SciPy contains functions not found in numpy.linalg, such as functions related to LU decomposition and the Schur decomposition, multiple ways of calculating the pseudoinverse, and matrix transcendentals such as the matrix logarithm. Some functions that exist in both have augmented functionality in scipy.linalg. For example, scipy.linalg.eig can take a second matrix argument for solving generalized eigenvalue problems. Some functions in NumPy, however, have more flexible broadcasting options. For example, numpy.linalg.solve can handle "stacked" arrays, while scipy.linalg.solve accepts only a single square array as its first argument.

.. note::

The term matrix as it is used on this page indicates a 2d numpy.array object, and not a numpy.matrix object. The latter is no longer recommended, even for linear algebra. See :ref:the matrix object documentation<matrix-objects> for more information.

The @ operator

Introduced in NumPy 1.10.0, the @ operator is preferable to other methods when computing the matrix product between 2d arrays. The :func:numpy.matmul function implements the @ operator.

.. currentmodule:: numpy

Matrix and vector products

.. autosummary:: :toctree: generated/

dot linalg.multi_dot vdot vecdot linalg.vecdot inner outer linalg.outer matmul linalg.matmul (Array API compatible location) matvec vecmat tensordot linalg.tensordot (Array API compatible location) einsum einsum_path linalg.matrix_power kron linalg.cross

Decompositions

.. autosummary:: :toctree: generated/

linalg.cholesky linalg.qr linalg.svd linalg.svdvals

Matrix eigenvalues

.. autosummary:: :toctree: generated/

linalg.eig linalg.eigh linalg.eigvals linalg.eigvalsh

Norms and other numbers

.. autosummary:: :toctree: generated/

linalg.norm linalg.matrix_norm (Array API compatible) linalg.vector_norm (Array API compatible) linalg.cond linalg.det linalg.matrix_rank linalg.slogdet trace linalg.trace (Array API compatible)

Solving equations and inverting matrices

.. autosummary:: :toctree: generated/

linalg.solve linalg.tensorsolve linalg.lstsq linalg.inv linalg.pinv linalg.tensorinv

Other matrix operations

.. autosummary:: :toctree: generated/

diagonal linalg.diagonal (Array API compatible) linalg.matrix_transpose (Array API compatible)

Exceptions

.. autosummary:: :toctree: generated/

linalg.LinAlgError

.. _routines.linalg-broadcasting:

Linear algebra on several matrices at once

Several of the linear algebra routines listed above are able to compute results for several matrices at once, if they are stacked into the same array.

This is indicated in the documentation via input parameter specifications such as a : (..., M, M) array_like. This means that if for instance given an input array a.shape == (N, M, M), it is interpreted as a "stack" of N matrices, each of size M-by-M. Similar specification applies to return values, for instance the determinant has det : (...) and will in this case return an array of shape det(a).shape == (N,). This generalizes to linear algebra operations on higher-dimensional arrays: the last 1 or 2 dimensions of a multidimensional array are interpreted as vectors or matrices, as appropriate for each operation.