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, subset_by_index=None, subset_by_value=None, driver=None)

Solves a standard or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

Find eigenvalues array w of array a, where b is positive definite such that for every eigenvalue λ (i-th entry of w) and its eigenvector vi (i-th column of v) satisfies:

              a @ vi = λ * b @ vi
vi.conj().T @ a @ vi = λ
vi.conj().T @ b @ vi = 1

In the standard problem, b is assumed to be the identity matrix.

Parameters

a(M, M) array_like

A complex Hermitian or real symmetric matrix whose eigenvalues 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.

lowerbool, optional

Whether the pertinent array data is taken from the lower or upper triangle of a and, if applicable, b. (Default: lower)

overwrite_abool, optional

Whether to overwrite data in a (may improve performance). Default is False.

overwrite_bbool, optional

Whether to overwrite data in b (may improve performance). Default is False.

typeint, optional

For the generalized problems, this keyword specifies the problem type to be solved for w and v (only takes 1, 2, 3 as possible inputs):

1 =>     a @ v = w @ b @ v
2 => a @ b @ v = w @ v
3 => b @ a @ v = w @ v

This keyword is ignored for standard problems.

check_finitebool, 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.

subset_by_indexiterable, optional

If provided, this two-element iterable defines the start and the end indices of the desired eigenvalues (ascending order and 0-indexed). To return only the second smallest to fifth smallest eigenvalues, [1, 4] is used. [n-3, n-1] returns the largest three. Only available with “evr”, “evx”, and “gvx” drivers. The entries are directly converted to integers via int().

subset_by_valueiterable, optional

If provided, this two-element iterable defines the half-open interval (a, b] that, if any, only the eigenvalues between these values are returned. Only available with “evr”, “evx”, and “gvx” drivers. Use np.inf for the unconstrained ends.

driverstr, optional

Defines which LAPACK driver should be used. Valid options are “ev”, “evd”, “evr”, “evx” for standard problems and “gv”, “gvd”, “gvx” for generalized (where b is not None) problems. See the Notes section of scipy.linalg.eigh.

turbobool, optional

Deprecated by ``driver=gvd`` option. Has no significant effect for eigenvalue computations since no eigenvectors are requested.

Deprecated since version 1.5.0.

eigvalstuple (lo, hi), optional

Deprecated by ``subset_by_index`` keyword. 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.

Deprecated since version 1.5.0.

Returns

w(N,) 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 will be reported but results will be wrong.

See also

eigh

eigenvalues and right eigenvectors for symmetric/Hermitian arrays

eigvals

eigenvalues of general arrays

eigvals_banded

eigenvalues for symmetric/Hermitian band matrices

eigvalsh_tridiagonal

eigenvalues of symmetric/Hermitian tridiagonal matrices

Notes

This function does not check the input array for being Hermitian/symmetric in order to allow for representing arrays with only their upper/lower triangular parts.

This function serves as a one-liner shorthand for scipy.linalg.eigh with the option eigvals_only=True to get the eigenvalues and not the eigenvectors. Here it is kept as a legacy convenience. It might be beneficial to use the main function to have full control and to be a bit more pythonic.

Examples

For more examples see scipy.linalg.eigh.

>>> from scipy.linalg import eigvalsh
>>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
>>> w = eigvalsh(A)
>>> w
array([-3.74637491, -0.76263923,  6.08502336, 12.42399079])