# jax.scipy.linalg.eigh¶

`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]

Solve a standard or generalized eigenvalue problem for a complex

LAX-backend implementation of `eigh()`.

Original docstring below.

Hermitian or real symmetric matrix.

Find eigenvalues array `w` and optionally eigenvectors array `v` 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 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` and, if applicable, `b`. (Default: lower)

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

• type (int, 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.

• overwrite_a (bool, optional) – Whether to overwrite data in `a` (may improve performance). Default is False.

• overwrite_b (bool, optional) – Whether to overwrite data in `b` (may improve performance). Default is False.

• 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.

• turbo (bool, optional) – Deprecated since v1.5.0, use ``driver=gvd`` keyword instead. Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if full set of eigenvalues are requested.). Has no significant effect if eigenvectors are not requested.

• eigvals (tuple (lo, hi), optional) – Deprecated since v1.5.0, use ``subset_by_index`` keyword instead. 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.

Returns

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

• v ((M, N) ndarray) – (if `eigvals_only == False`)