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# torch.linalg.eigvalsh

torch.linalg.eigvalsh(A, UPLO='L', *, out=None) → Tensor

  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∈K n×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−λ I n​)λ∈R

  where I n\mathrm{I}_n I n​ 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.

  See also

  `torch.linalg.eigh()` computes the full eigenvalue decomposition.

  Parameters:

    - 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’.

  Keyword Arguments:

    out (Tensor, optional) – output tensor. Ignored if None. Default: None.

  Returns:

    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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