# 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]) 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, 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_triangular(a, b[, trans, lower, ...]) Solve the equation a x = b for x, assuming a is a triangular matrix. det(a[, overwrite_a]) Compute the determinant of a matrix norm(a[, ord]) Matrix or vector norm. lstsq(a, b[, cond, overwrite_a, overwrite_b]) Compute least-squares solution to equation Ax = b. pinv(a[, cond, rcond]) Compute the (Moore-Penrose) pseudo-inverse of a matrix. pinv2(a[, cond, rcond]) Compute the (Moore-Penrose) pseudo-inverse of a matrix. kron(a, b) Kronecker product of a and b. tril(m[, k]) Construct a copy of a matrix with elements above the k-th diagonal zeroed. triu(m[, k]) Construct a copy of a matrix with elements below the k-th diagonal zeroed.

## 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]) 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 eigvalsh(a[, b, lower, overwrite_a, ...]) Solve an ordinary or generalized eigenvalue problem for a complex 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]) Compute pivoted LU decompostion of a matrix. lu_factor(a[, overwrite_a]) Compute pivoted LU decomposition of a matrix. lu_solve((lu, piv), b[, trans, overwrite_b]) Solve an equation system, a x = b, given the LU factorization of a svd(a[, full_matrices, compute_uv, overwrite_a]) Singular Value Decomposition. svdvals(a[, overwrite_a]) 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]) 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]) Compute the Cholesky decomposition of a matrix, to use in cho_solve cho_solve((c, lower), b[, overwrite_b]) Solve the linear equations A x = b, given the Cholesky factorization of A. cho_solve_banded((cb, lower), b[, overwrite_b]) Solve the linear equations A x = b, given the Cholesky factorization of A. qr(a[, overwrite_a, lwork, mode, pivoting]) Compute QR decomposition of a matrix. schur(a[, output, lwork, overwrite_a, sort]) Compute Schur decomposition of a matrix. rsf2csf(T, Z) Convert real Schur form to complex Schur form. hessenberg(a[, calc_q, overwrite_a]) Compute Hessenberg form of a matrix.

## Matrix Functions¶

 expm(A[, q]) Compute the matrix exponential using Pade approximation. expm2(A) Compute the matrix exponential using eigenvalue decomposition. expm3(A[, q]) Compute the matrix exponential using Taylor series. 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]) Matrix square root. funm(A, func[, disp]) Evaluate a matrix function specified by a callable.

## 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. hadamard(n[, dtype]) Construct a Hadamard matrix. hankel(c[, r]) Construct a Hankel matrix. 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. 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.