# Linear algebra (scipy.linalg)¶

Linear algebra functions.

numpy.linalg 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¶

 inv(a[, overwrite_a, check_finite]) Compute the inverse of a matrix. solve(a, b[, sym_pos, lower, overwrite_a, ...]) Solve the equation a x = b for x. solve_banded(l_and_u, ab, b[, overwrite_ab, ...]) Solve the equation a x = b for x, assuming a is banded matrix. solveh_banded(ab, b[, overwrite_ab, ...]) Solve equation a x = b. solve_circulant(c, b[, singular, tol, ...]) Solve C x = b for x, where C is a circulant matrix. solve_triangular(a, b[, trans, lower, ...]) Solve the equation a x = b for x, assuming a is a triangular matrix. solve_toeplitz(c_or_cr, b[, check_finite]) Solve a Toeplitz system using Levinson Recursion det(a[, overwrite_a, check_finite]) Compute the determinant of a matrix norm(a[, ord, axis, keepdims]) Matrix or vector norm. lstsq(a, b[, cond, overwrite_a, ...]) Compute least-squares solution to equation Ax = b. pinv(a[, cond, rcond, return_rank, check_finite]) Compute the (Moore-Penrose) pseudo-inverse of a matrix. pinv2(a[, cond, rcond, return_rank, ...]) Compute the (Moore-Penrose) pseudo-inverse of a matrix. pinvh(a[, cond, rcond, lower, return_rank, ...]) Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix. kron(a, b) Kronecker product. tril(m[, k]) Make a copy of a matrix with elements above the k-th diagonal zeroed. triu(m[, k]) Make a copy of a matrix with elements below the k-th diagonal zeroed. orthogonal_procrustes(A, B[, check_finite]) Compute the matrix solution of the orthogonal Procrustes problem. LinAlgError Generic Python-exception-derived object raised by linalg functions.

## Eigenvalue Problems¶

 eig(a[, b, left, right, overwrite_a, ...]) Solve an ordinary or generalized eigenvalue problem of a square matrix. eigvals(a[, b, overwrite_a, check_finite]) Compute eigenvalues from an ordinary or generalized eigenvalue problem. eigh(a[, b, lower, eigvals_only, ...]) Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix. eigvalsh(a[, b, lower, overwrite_a, ...]) Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix. eig_banded(a_band[, lower, eigvals_only, ...]) Solve real symmetric or complex hermitian band matrix eigenvalue problem. eigvals_banded(a_band[, lower, ...]) Solve real symmetric or complex hermitian band matrix eigenvalue problem.

## Decompositions¶

 lu(a[, permute_l, overwrite_a, check_finite]) Compute pivoted LU decomposition of a matrix. lu_factor(a[, overwrite_a, check_finite]) Compute pivoted LU decomposition of a matrix. lu_solve(lu_and_piv, b[, trans, ...]) Solve an equation system, a x = b, given the LU factorization of a svd(a[, full_matrices, compute_uv, ...]) Singular Value Decomposition. svdvals(a[, overwrite_a, check_finite]) Compute singular values of a matrix. diagsvd(s, M, N) Construct the sigma matrix in SVD from singular values and size M, N. orth(A) Construct an orthonormal basis for the range of A using SVD cholesky(a[, lower, overwrite_a, check_finite]) Compute the Cholesky decomposition of a matrix. cholesky_banded(ab[, overwrite_ab, lower, ...]) Cholesky decompose a banded Hermitian positive-definite matrix cho_factor(a[, lower, overwrite_a, check_finite]) Compute the Cholesky decomposition of a matrix, to use in cho_solve cho_solve(c_and_lower, b[, overwrite_b, ...]) Solve the linear equations A x = b, given the Cholesky factorization of A. cho_solve_banded(cb_and_lower, b[, ...]) Solve the linear equations A x = b, given the Cholesky factorization of A. polar(a[, side]) Compute the polar decomposition. qr(a[, overwrite_a, lwork, mode, pivoting, ...]) Compute QR decomposition of a matrix. qr_multiply(a, c[, mode, pivoting, ...]) Calculate the QR decomposition and multiply Q with a matrix. qr_update(Q, R, u, v[, overwrite_qruv, ...]) Rank-k QR update qr_delete(Q, R, k[, p, which, overwrite_qr, ...]) QR downdate on row or column deletions qr_insert(Q, R, u, k[, which, rcond, ...]) QR update on row or column insertions rq(a[, overwrite_a, lwork, mode, check_finite]) Compute RQ decomposition of a matrix. qz(A, B[, output, lwork, sort, overwrite_a, ...]) QZ decomposition for generalized eigenvalues of a pair of matrices. ordqz(A, B[, sort, output, overwrite_a, ...]) QZ decomposition for a pair of matrices with reordering. schur(a[, output, lwork, overwrite_a, sort, ...]) Compute Schur decomposition of a matrix. rsf2csf(T, Z[, check_finite]) Convert real Schur form to complex Schur form. hessenberg(a[, calc_q, overwrite_a, ...]) Compute Hessenberg form of a matrix.

scipy.linalg.interpolative – Interpolative matrix decompositions

## Matrix Functions¶

 expm(A[, q]) Compute the matrix exponential using Pade approximation. logm(A[, disp]) Compute matrix logarithm. cosm(A) Compute the matrix cosine. sinm(A) Compute the matrix sine. tanm(A) Compute the matrix tangent. coshm(A) Compute the hyperbolic matrix cosine. sinhm(A) Compute the hyperbolic matrix sine. tanhm(A) Compute the hyperbolic matrix tangent. signm(A[, disp]) Matrix sign function. sqrtm(A[, disp, blocksize]) Matrix square root. funm(A, func[, disp]) Evaluate a matrix function specified by a callable. expm_frechet(A, E[, method, compute_expm, ...]) Frechet derivative of the matrix exponential of A in the direction E. expm_cond(A[, check_finite]) Relative condition number of the matrix exponential in the Frobenius norm. fractional_matrix_power(A, t) Compute the fractional power of a matrix.

## Matrix Equation Solvers¶

 solve_sylvester(a, b, q) Computes a solution (X) to the Sylvester equation $$AX + XB = Q$$. solve_continuous_are(a, b, q, r) Solves the continuous algebraic Riccati equation (CARE). solve_discrete_are(a, b, q, r) Solves the discrete algebraic Riccati equation (DARE). solve_discrete_lyapunov(a, q[, method]) Solves the discrete Lyapunov equation $$A'XA-X=-Q$$. solve_lyapunov(a, q) Solves the continuous Lyapunov equation $$AX + XA^H = Q$$.

## Special Matrices¶

 block_diag(*arrs) Create a block diagonal matrix from provided arrays. circulant(c) Construct a circulant matrix. companion(a) Create a companion matrix. dft(n[, scale]) Discrete Fourier transform matrix. hadamard(n[, dtype]) Construct a Hadamard matrix. hankel(c[, r]) Construct a Hankel matrix. helmert(n[, full]) Create a Helmert matrix of order n. hilbert(n) Create a Hilbert matrix of order n. invhilbert(n[, exact]) Compute the inverse of the Hilbert matrix of order n. leslie(f, s) Create a Leslie matrix. pascal(n[, kind, exact]) Returns the n x n Pascal matrix. invpascal(n[, kind, exact]) Returns the inverse of the n x n Pascal matrix. toeplitz(c[, r]) Construct a Toeplitz matrix. tri(N[, M, k, dtype]) Construct (N, M) matrix filled with ones at and below the k-th diagonal.

## Low-level routines¶

 get_blas_funcs(names[, arrays, dtype]) Return available BLAS function objects from names. get_lapack_funcs(names[, arrays, dtype]) Return available LAPACK function objects from names. find_best_blas_type([arrays, dtype]) Find best-matching BLAS/LAPACK type.

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