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# Linear algebra (`scipy.linalg`)

Linear algebra functions.

See also

numpy.linalg for more linear algebra functions. Note that 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[, lower, overwrite_a, ...]) | Solve the equation `a @ x = b` for `x`, where a is a square matrix.
`solve_banded`(l_and_u, ab, b[, overwrite_ab, ...]) | Solve the equation `a @ x = b` for `x`, where `a` is the banded matrix defined by ab.
`solveh_banded`(ab, b[, overwrite_ab, ...]) | Solve the equation `a @ x = b` for `x`, where `a` is the Hermitian positive-definite banded matrix defined by ab.
`solve_circulant`(c, b[, singular, tol, ...]) | Solve the equation `C @ x = b` for `x`, where `C` is a circulant matrix defined by c.
`solve_triangular`(a, b[, trans, lower, ...]) | Solve the equation `a @ x = b` for `x`, where a is a triangular matrix.
`solve_toeplitz`(c_or_cr, b[, check_finite]) | Solve the equation `T @ x = b` for `x`, where `T` is a Toeplitz matrix defined by c_or_cr.
`matmul_toeplitz`(c_or_cr, x[, check_finite, ...]) | Efficient Toeplitz Matrix-Matrix Multiplication using FFT.
`det`(a[, overwrite_a, check_finite]) | Compute the determinant of a matrix.
`norm`(a[, ord, axis, keepdims, check_finite]) | Matrix or vector norm.
`lstsq`(a, b[, cond, overwrite_a, ...]) | Compute least-squares solution to the equation `a @ x = b`.
`pinv`(a, *[, atol, rtol, return_rank, ...]) | Compute the (Moore-Penrose) pseudo-inverse of a matrix.
`pinvh`(a[, atol, rtol, lower, return_rank, ...]) | Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.
`khatri_rao`(a, b) | Khatri-Rao product of two matrices.
`orthogonal_procrustes`(A, B[, check_finite]) | Compute the matrix solution of the orthogonal (or unitary) Procrustes problem.
`matrix_balance`(A[, permute, scale, ...]) | Compute a diagonal similarity transformation for row/column balancing.
`subspace_angles`(A, B) | Compute the subspace angles between two matrices.
`bandwidth`(a) | Return the lower and upper bandwidth of a numeric array.
`issymmetric`(a[, atol, rtol]) | Check if a square 2D array is symmetric.
`ishermitian`(a[, atol, rtol]) | Check if a square 2D array is Hermitian.
`LinAlgError` | Generic Python-exception-derived object raised by linalg functions.
`LinAlgWarning` | The warning emitted when a linear algebra related operation is close to fail conditions of the algorithm or loss of accuracy is expected.

## 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, overwrite_b, ...]) | Compute eigenvalues from an ordinary or generalized eigenvalue problem.
`eigh`(a[, b, lower, eigvals_only, ...]) | Solve a standard or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.
`eigvalsh`(a[, b, lower, overwrite_a, ...]) | Solves a standard 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.
`eigh_tridiagonal`(d, e[, eigvals_only, ...]) | Solve eigenvalue problem for a real symmetric tridiagonal matrix.
`eigvalsh_tridiagonal`(d, e[, select, ...]) | Solve eigenvalue problem for a real symmetric tridiagonal matrix.

## Decompositions

`lu`(a[, permute_l, overwrite_a, ...]) | Compute LU decomposition of a matrix with partial pivoting.
`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[, rcond]) | Construct an orthonormal basis for the range of A using SVD.
`null_space`(A[, rcond, overwrite_a, ...]) | Construct an orthonormal basis for the null space of A using SVD.
`ldl`(A[, lower, hermitian, overwrite_a, ...]) | Computes the LDLt or Bunch-Kaufman factorization of a symmetric/ hermitian matrix.
`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 the banded Hermitian `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, int p=1[, which, ...]) | 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.
`cdf2rdf`(w, v) | Complex diagonal form to real diagonal block form.
`cossin`(X[, p, q, separate, swap_sign, ...]) | Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix.

See also

`scipy.linalg.interpolative` – Interpolative matrix decompositions

## Matrix Functions

`expm`(A) | Compute the matrix exponential of an array.
`logm`(A) | 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) | Matrix sign function.
`sqrtm`(A) | Compute, if exists, the 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[, e, s, ...]) | Solves the continuous-time algebraic Riccati equation (CARE).
`solve_discrete_are`(a, b, q, r[, e, s, balanced]) | Solves the discrete-time algebraic Riccati equation (DARE).
`solve_continuous_lyapunov`(a, q) | Solves the continuous Lyapunov equation \(AX + XA^H = Q\).
`solve_discrete_lyapunov`(a, q[, method]) | Solves the discrete Lyapunov equation \(AXA^H - X + Q = 0\).

## Sketches and Random Projections

`clarkson_woodruff_transform`(input_matrix, ...) | Applies a Clarkson-Woodruff Transform/sketch to the input matrix.

## Special Matrices

`block_diag`(*arrs) | Create a block diagonal array from provided arrays.
`circulant`(c) | Construct a circulant matrix.
`companion`(a) | Create a companion matrix.
`convolution_matrix`(a, n[, mode]) | Construct a convolution matrix.
`dft`(n[, scale]) | Discrete Fourier transform matrix.
`fiedler`(a) | Returns a symmetric Fiedler matrix.
`fiedler_companion`(a) | Returns a Fiedler companion 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.

## Low-level routines

`get_blas_funcs`(names[, arrays, dtype, ilp64]) | Return available BLAS function objects from names.
`get_lapack_funcs`(names[, arrays, dtype, ilp64]) | Return available LAPACK function objects from names.
`find_best_blas_type`([arrays, dtype]) | Find best-matching BLAS/LAPACK type.

See also

`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