reference:routines.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.

The SciPy library also contains a 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.

The @ operator

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

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

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.