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====================================
Linear algebra (:mod:`scipy.linalg`)
====================================
.. currentmodule:: scipy.linalg
Linear algebra functions.
.. eventually, we should replace the numpy.linalg HTML link with just `numpy.linalg`
.. seealso::
`numpy.linalg <https://www.numpy.org/devdocs/reference/routines.linalg.html>`__
for more linear algebra functions. Note that
although `scipy.linalg` imports most of them, identically named
functions from `scipy.linalg` may offer more or slightly differing
functionality.
Basics
======
.. autosummary::
:toctree: generated/
inv - Find the inverse of a square matrix
solve - Solve a linear system of equations
solve_banded - Solve a banded linear system
solveh_banded - Solve a Hermitian or symmetric banded system
solve_circulant - Solve a circulant system
solve_triangular - Solve a triangular matrix
solve_toeplitz - Solve a toeplitz matrix
matmul_toeplitz - Multiply a Toeplitz matrix with an array.
det - Find the determinant of a square matrix
norm - Matrix and vector norm
lstsq - Solve a linear least-squares problem
pinv - Pseudo-inverse (Moore-Penrose) using lstsq
pinvh - Pseudo-inverse of hermitian matrix
kron - Kronecker product of two arrays
khatri_rao - Khatri-Rao product of two arrays
tril - Construct a lower-triangular matrix from a given matrix
triu - Construct an upper-triangular matrix from a given matrix
orthogonal_procrustes - Solve an orthogonal Procrustes problem
matrix_balance - Balance matrix entries with a similarity transformation
subspace_angles - Compute the subspace angles between two matrices
bandwidth - Return the lower and upper bandwidth of an array
issymmetric - Check if a square 2D array is symmetric
ishermitian - Check if a square 2D array is Hermitian
LinAlgError
LinAlgWarning
Eigenvalue Problems
===================
.. autosummary::
:toctree: generated/
eig - Find the eigenvalues and eigenvectors of a square matrix
eigvals - Find just the eigenvalues of a square matrix
eigh - Find the e-vals and e-vectors of a Hermitian or symmetric matrix
eigvalsh - Find just the eigenvalues of a Hermitian or symmetric matrix
eig_banded - Find the eigenvalues and eigenvectors of a banded matrix
eigvals_banded - Find just the eigenvalues of a banded matrix
eigh_tridiagonal - Find the eigenvalues and eigenvectors of a tridiagonal matrix
eigvalsh_tridiagonal - Find just the eigenvalues of a tridiagonal matrix
Decompositions
==============
.. autosummary::
:toctree: generated/
lu - LU decomposition of a matrix
lu_factor - LU decomposition returning unordered matrix and pivots
lu_solve - Solve Ax=b using back substitution with output of lu_factor
svd - Singular value decomposition of a matrix
svdvals - Singular values of a matrix
diagsvd - Construct matrix of singular values from output of svd
orth - Construct orthonormal basis for the range of A using svd
null_space - Construct orthonormal basis for the null space of A using svd
ldl - LDL.T decomposition of a Hermitian or a symmetric matrix.
cholesky - Cholesky decomposition of a matrix
cholesky_banded - Cholesky decomp. of a sym. or Hermitian banded matrix
cho_factor - Cholesky decomposition for use in solving a linear system
cho_solve - Solve previously factored linear system
cho_solve_banded - Solve previously factored banded linear system
polar - Compute the polar decomposition.
qr - QR decomposition of a matrix
qr_multiply - QR decomposition and multiplication by Q
qr_update - Rank k QR update
qr_delete - QR downdate on row or column deletion
qr_insert - QR update on row or column insertion
rq - RQ decomposition of a matrix
qz - QZ decomposition of a pair of matrices
ordqz - QZ decomposition of a pair of matrices with reordering
schur - Schur decomposition of a matrix
rsf2csf - Real to complex Schur form
hessenberg - Hessenberg form of a matrix
cdf2rdf - Complex diagonal form to real diagonal block form
cossin - Cosine sine decomposition of a unitary or orthogonal matrix
.. seealso::
`scipy.linalg.interpolative` -- Interpolative matrix decompositions
Matrix Functions
================
.. autosummary::
:toctree: generated/
expm - Matrix exponential
logm - Matrix logarithm
cosm - Matrix cosine
sinm - Matrix sine
tanm - Matrix tangent
coshm - Matrix hyperbolic cosine
sinhm - Matrix hyperbolic sine
tanhm - Matrix hyperbolic tangent
signm - Matrix sign
sqrtm - Matrix square root
funm - Evaluating an arbitrary matrix function
expm_frechet - Frechet derivative of the matrix exponential
expm_cond - Relative condition number of expm in the Frobenius norm
fractional_matrix_power - Fractional matrix power
Matrix Equation Solvers
=======================
.. autosummary::
:toctree: generated/
solve_sylvester - Solve the Sylvester matrix equation
solve_continuous_are - Solve the continuous-time algebraic Riccati equation
solve_discrete_are - Solve the discrete-time algebraic Riccati equation
solve_continuous_lyapunov - Solve the continuous-time Lyapunov equation
solve_discrete_lyapunov - Solve the discrete-time Lyapunov equation
Sketches and Random Projections
===============================
.. autosummary::
:toctree: generated/
clarkson_woodruff_transform - Applies the Clarkson Woodruff Sketch (a.k.a CountMin Sketch)
Special Matrices
================
.. autosummary::
:toctree: generated/
block_diag - Construct a block diagonal matrix from submatrices
circulant - Circulant matrix
companion - Companion matrix
convolution_matrix - Convolution matrix
dft - Discrete Fourier transform matrix
fiedler - Fiedler matrix
fiedler_companion - Fiedler companion matrix
hadamard - Hadamard matrix of order 2**n
hankel - Hankel matrix
helmert - Helmert matrix
hilbert - Hilbert matrix
invhilbert - Inverse Hilbert matrix
leslie - Leslie matrix
pascal - Pascal matrix
invpascal - Inverse Pascal matrix
toeplitz - Toeplitz matrix
tri - Construct a matrix filled with ones at and below a given diagonal
Low-level routines
==================
.. autosummary::
:toctree: generated/
get_blas_funcs
get_lapack_funcs
find_best_blas_type
.. seealso::
`scipy.linalg.blas` -- Low-level BLAS functions
`scipy.linalg.lapack` -- Low-level LAPACK functions
`scipy.linalg.cython_blas` -- Low-level BLAS functions for Cython
`scipy.linalg.cython_lapack` -- Low-level LAPACK functions for Cython
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