jax.lax.linalg.eigh

jax.lax.linalg.eigh(x, *, lower=True, symmetrize_input=True, sort_eigenvalues=True, subset_by_index=None)

Eigendecomposition of a Hermitian matrix.

Computes the eigenvectors and eigenvalues of a complex Hermitian or real symmetric square matrix.

Parameters:

• x (Array) – A batch of square complex Hermitian or real symmetric matrices with shape [..., n, n].

• lower (bool) – If symmetrize_input is False, describes which triangle of the input matrix to use. If symmetrize_input is False, only the triangle given by lower is accessed; the other triangle is ignored and not accessed.

• symmetrize_input (bool) – If True, the matrix is symmetrized before the eigendecomposition by computing \(\frac{1}{2}(x + x^H)\).

• sort_eigenvalues (bool) –

  If True, the eigenvalues will be sorted in ascending

  order. If False the eigenvalues are returned in an implementation-defined order.

  subset_by_index: Optional 2-tuple [start, end] indicating the range of

  indices of eigenvalues to compute. For example, is range_select = [n-2,n], then eigh computes the two largest eigenvalues and their eigenvectors.

• subset_by_index (tuple[int, int] | None)

Return type:

tuple[Array, Array]

Returns:

A tuple (v, w).

v is an array with the same dtype as x such that v[..., :, i] is the normalized eigenvector corresponding to eigenvalue w[..., i].

w is an array with the same dtype as x (or its real counterpart if complex) with shape [..., d] containing the eigenvalues of x in ascending order(each repeated according to its multiplicity). If subset_by_index is None then d is equal to n. Otherwise d is equal to subset_by_index[1] - subset_by_index[0].