Letting be or , the eigenvalue decomposition of a complex Hermitian or real symmetric matrix 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'.

## Value

A list (eigenvalues, eigenvectors) which corresponds to and 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.

## Details

$A = Q \operatorname{diag}(\Lambda) Q^{H}\mathrlap{\qquad Q \in \mathbb{K}^{n \times n}, \Lambda \in \mathbb{R}^n}$

where is the conjugate transpose when is complex, and the transpose when is real-valued. 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.

## Note

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

## 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 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 through the computation of .

• 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)
}
#> [[1]]
#> torch_tensor
#>  0.6851
#>  2.1951
#> [ CPUFloatType{2} ]
#>
#> [[2]]
#> torch_tensor
#>  0.5609 -0.8279
#> -0.8279 -0.5609
#> [ CPUFloatType{2,2} ]
#>