linalg_eigh {torch}

Computes the eigenvalue decomposition of a complex Hermitian or real symmetric matrix.

Description

Letting \mathbb{K} be \mathbb{R} or \mathbb{C}, the eigenvalue decomposition of a complex Hermitian or real symmetric matrix A \in \mathbb{K}^{n \times n} is defined as

Usage

linalg_eigh(A, UPLO = "L")

Arguments

A

(Tensor): tensor of shape ⁠(*, n, n)⁠ where * is zero or more batch dimensions consisting of symmetric or Hermitian matrices.

UPLO

('L', 'U', optional): controls whether to use the upper or lower triangular part of A in the computations. Default: 'L'.

Details

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where Q^{H} is the conjugate transpose when Q is complex, and the transpose when Q is real-valued. Q is orthogonal in the real case and unitary in the complex case.

Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A is a batch of matrices then the output has the same batch dimensions.

A is assumed to be Hermitian (resp. symmetric), but this is not checked internally, instead:

• If UPLO\ ⁠= 'L'⁠ (default), only the lower triangular part of the matrix is used in the computation.

• If UPLO\ ⁠= 'U'⁠, only the upper triangular part of the matrix is used. The eigenvalues are returned in ascending order.

Value

A list ⁠(eigenvalues, eigenvectors)⁠ which corresponds to \Lambda and Q above. eigenvalues will always be real-valued, even when A is complex.

It will also be ordered in ascending order. eigenvectors will have the same dtype as A and will contain the eigenvectors as its columns.

Warning

• The eigenvectors of a symmetric matrix are not unique, nor are they continuous with respect to A. Due to this lack of uniqueness, different hardware and software may compute different eigenvectors. This non-uniqueness is caused by the fact that multiplying an eigenvector by -1 in the real case or by e^{i \phi}, \phi \in \mathbb{R} in the complex case produces another set of valid eigenvectors of the matrix. This non-uniqueness problem is even worse when the matrix has repeated eigenvalues. In this case, one may multiply the associated eigenvectors spanning the subspace by a rotation matrix and the resulting eigenvectors will be valid eigenvectors.

• Gradients computed using the eigenvectors tensor will only be finite when A has unique eigenvalues. Furthermore, if the distance between any two eigvalues is close to zero, the gradient will be numerically unstable, as it depends on the eigenvalues \lambda_i through the computation of \frac{1}{\min_{i \neq j} \lambda_i - \lambda_j}.

Note

The eigenvalues of real symmetric or complex Hermitian matrices are always real.

See Also

• linalg_eigvalsh() computes only the eigenvalues values of a Hermitian matrix. Unlike linalg_eigh(), the gradients of linalg_eigvalsh() are always numerically stable.

• linalg_cholesky() for a different decomposition of a Hermitian matrix. The Cholesky decomposition gives less information about the matrix but is much faster to compute than the eigenvalue decomposition.

• linalg_eig() for a (slower) function that computes the eigenvalue decomposition of a not necessarily Hermitian square matrix.

• linalg_svd() for a (slower) function that computes the more general SVD decomposition of matrices of any shape.

• linalg_qr() for another (much faster) decomposition that works on general matrices.

Other linalg: linalg_cholesky_ex(), linalg_cholesky(), linalg_det(), linalg_eigvalsh(), linalg_eigvals(), linalg_eig(), linalg_householder_product(), linalg_inv_ex(), linalg_inv(), linalg_lstsq(), linalg_matrix_norm(), linalg_matrix_power(), linalg_matrix_rank(), linalg_multi_dot(), linalg_norm(), linalg_pinv(), linalg_qr(), linalg_slogdet(), linalg_solve_triangular(), linalg_solve(), linalg_svdvals(), linalg_svd(), linalg_tensorinv(), linalg_tensorsolve(), linalg_vector_norm()

Examples

if (torch_is_installed()) {
a <- torch_randn(2, 2)
linalg_eigh(a)
}