heat.linalg.eigh

Implements Symmetric Eigenvalue Decomposition

Module Contents

eigh(A: heat.core.dndarray.DNDarray, r_max_zolopd: int = 8, silent: bool = True) → Tuple[heat.core.dndarray.DNDarray, heat.core.dndarray.DNDarray][source]

Computes the symmetric eigenvalue decomposition of a symmetric n x n - matrix A, provided as a DNDarray.

The function returns DNDarrays Lambda (shape (n,) with split = 0) and V (shape (n,n)) such that A = V @ diag(Lambda) @ V^T, where Lambda contains the eigenvalues of A and V is an orthonormal matrix containing the corresponding eigenvectors as columns.

Parameters:

  • A (DNDarray) – The input matrix. Must be symmetric.

  • r_max_zolopd (int, optional) – This is a hyperparameter for the computation of the polar decomposition via heat.linalg.polar() which is applied multiple times in this function. See the documentation of heat.linalg.polar() for more details on its meaning and the respective default value.

  • silent (bool, optional) – If True (default), suppresses output messages; otherwise, some information on the recursion is printed to the console.

Notes

Unlike the torch.linalg.eigh() function, the eigenvalues are returned in descending order. Note that no check of symmetry is performed on the input matrix A; thus, applying this function to a non-symmetric matrix may result in unpredictable behaviour without a specific error message pointing to this issue.

The algorithm used for the computation of the symmetric eigenvalue decomposition is based on the Zolotarev polar decomposition; see Algorithm 5.2 in:

Nakatsukasa, Y., & Freund, R. W. (2016). Computing fundamental matrix decompositions accurately via the matrix sign function in two iterations: The power of Zolotarev’s functions. SIAM Review, 58(3).

See also

heat.linalg.polar()