scipy.linalg.eigvalsh¶

scipy.linalg.eigvalsh(a, b=None, lower=True, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1, check_finite=True)[source]¶

Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

Find eigenvalues w of matrix a, where b is positive definite:

              a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1

Parameters:

Parameters:	a : (M, M) array_like A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed. lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower) turbo : bool, optional Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None) eigvals : tuple (lo, hi), optional Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned. type : int, optional Specifies the problem type to be solved: type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i] = w[i] v[:,i] overwrite_a : bool, optional Whether to overwrite data in a (may improve performance) overwrite_b : bool, optional Whether to overwrite data in b (may improve performance) check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns:	w : (N,) float ndarray The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.
Raises:	LinAlgError : If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or hermitian, no error is reported but results will be wrong.

a : (M, M) array_like

A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed.

b : (M, M) array_like, optional

A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.

lower : bool, optional

Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)

turbo : bool, optional

Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)

eigvals : tuple (lo, hi), optional

Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned.

type : int, optional

Specifies the problem type to be solved:

type = 1: a v[:,i] = w[i] b v[:,i]

type = 2: a b v[:,i] = w[i] v[:,i]

type = 3: b a v[:,i] = w[i] v[:,i]

overwrite_a : bool, optional

Whether to overwrite data in a (may improve performance)

overwrite_b : bool, optional

Whether to overwrite data in b (may improve performance)

check_finite : bool, optional

Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns:

w : (N,) float ndarray

The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.

Raises:

LinAlgError :

If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or hermitian, no error is reported but results will be wrong.