scipy.linalg.eigh

scipy.linalg.eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1, check_finite=True)

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

Find eigenvalues w and optionally eigenvectors v 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:

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)

eigvals_only : bool, optional

Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated)

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.

v : (M, N) complex ndarray

(if eigvals_only == False)

The normalized selected eigenvector corresponding to the eigenvalue w[i] is the column v[:,i].

Normalization:

type 1 and 3: v.conj() a v = w

type 2: inv(v).conj() a inv(v) = w

type = 1 or 2: v.conj() b v = I

type = 3: v.conj() inv(b) v = I

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.

See also

eigvalsh

eigenvalues of symmetric or Hermitian arrays

eig

eigenvalues and right eigenvectors for non-symmetric arrays

eigh

eigenvalues and right eigenvectors for symmetric/Hermitian arrays

eigh_tridiagonal

eigenvalues and right eiegenvectors for 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.

Examples

>>> from scipy.linalg import eigh
>>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
>>> w, v = eigh(A)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True