# jax.scipy.linalg.eighΒΆ

jax.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]ΒΆ
Solve an ordinary or generalized eigenvalue problem for a complex

Hermitian or real symmetric matrix.

LAX-backend implementation of eigh(). Original docstring below.

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) β A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed.

• b ((M, M) array_like, optional) β A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.

• lower (bool, optional) β Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)

• eigvals_only (bool, optional) β Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated)

• turbo (bool, optional) β Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)

• eigvals (tuple (lo, hi), optional) β 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) β Specifies the problem type to be solved:

• overwrite_a (bool, optional) β Whether to overwrite data in a (may improve performance)

• overwrite_b (bool, optional) β Whether to overwrite data in b (may improve performance)

• check_finite (bool, optional) β 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) β The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.

• v ((M, N) complex ndarray) β (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 β 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.

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