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)[source]¶

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:

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.

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.