eigh

→Tensor

eigh(x: Tensor, UPLO: str = 'L')

source

Eigendecomposition of a Hermitian / symmetric matrix.

Returns the eigenvalues and orthonormal eigenvectors of a real symmetric (or complex Hermitian) matrix $A$ . Eigenvalues are returned in ascending order and eigenvectors form an orthogonal matrix $V$ such that

A \,=\, V \,\Lambda\, V^\top,

where $\Lambda = \mathrm{diag}(\lambda_1, \ldots, \lambda_n)$ with $\lambda_1 \le \cdots \le \lambda_n$ .

Parameters

xTensor

Square Hermitian / symmetric matrix of shape (*, n, n).

UPLOstr= 'L'

"L" (default) reads only the lower triangle of x; "U" reads only the upper triangle. The other triangle is ignored, so a non-Hermitian input is accepted as long as the chosen triangle holds the correct values.

Returns

Tensor

Real-valued tensor of shape (*, n) in ascending order.

Notes

Prefer this over eig whenever A is symmetric / Hermitian — it is faster, numerically more stable, and guarantees real eigenvalues with orthogonal eigenvectors ( $V^\top V = I$ ).

Backward uses the Loewner-matrix formula $F_{ij} = 1/(\lambda_i - \lambda_j)$ :

\frac{\partial \mathcal{L}}{\partial A} \,=\, V\,\big(\mathrm{diag}(G_\lambda) + F \odot (V^\top G_V)\big)\,V^\top,

symmetrised to enforce the symmetry of $A$ . Gradients blow up near repeated eigenvalues.

Implementation: LAPACK syevd on the CPU stream (via Apple Accelerate); MLX on the GPU stream. Both run in $O(n^3)$ .

Examples

>>> import lucid
>>> from lucid.linalg import eigh
>>> A = lucid.tensor([[2.0, 1.0], [1.0, 3.0]])
>>> w, V = eigh(A)
>>> w
Tensor([1.3820, 3.6180])