Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
Letting K \mathbb{K} K be R \mathbb{R} R or C \mathbb{C} C, the eigenvalues of a complex Hermitian or real symmetric matrix A ∈ K n × n A \in \mathbb{K}^{n \times n} A∈Kn×n are defined as the roots (counted with multiplicity) of the polynomial p of degree n given by
p ( λ ) = det ( A − λ I n ) λ ∈ R p(\lambda) = \operatorname{det}(A - \lambda \mathrm{I}_n)\mathrlap{\qquad \lambda \in \mathbb{R}} p(λ)=det(A−λIn)λ∈R
where I n \mathrm{I}_n In is the n-dimensional identity matrix. The eigenvalues of a real symmetric or complex Hermitian matrix are always real.
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.
The eigenvalues are returned in ascending order.
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.
Note
When inputs are on a CUDA device, this function synchronizes that device with the CPU.
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’.
out (Tensor, optional) – output tensor. Ignored if None. Default: None.
A real-valued tensor containing the eigenvalues even when A is complex. The eigenvalues are returned in ascending order.
Examples:
>>> A = torch.randn(2, 2, dtype=torch.complex128)
>>> A = A + A.T.conj() # creates a Hermitian matrix
>>> A
tensor([[2.9228+0.0000j, 0.2029-0.0862j],
[0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)
>>> torch.linalg.eigvalsh(A)
tensor([0.3277, 2.9415], dtype=torch.float64)
>>> A = torch.randn(3, 2, 2, dtype=torch.float64)
>>> A = A + A.mT # creates a batch of symmetric matrices
>>> torch.linalg.eigvalsh(A)
tensor([[ 2.5797, 3.4629],
[-4.1605, 1.3780],
[-3.1113, 2.7381]], dtype=torch.float64)
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