torch.linalg.eigh

torch.linalg.eigh(A, UPLO='L', *, out=None)

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

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

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

where $Q^{\text{H}}$ 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.

Note

When inputs are on a CUDA device, this function synchronizes that device with the CPU.

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 $e^{i \phi}, \phi \in \mathbb{R}$ 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.

Warning

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

See also

torch.linalg.eigvalsh() computes only the eigenvalues of a Hermitian matrix. Unlike torch.linalg.eigh(), the gradients of eigvalsh() are always numerically stable.

torch.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.

torch.linalg.eig() for a (slower) function that computes the eigenvalue decomposition of a not necessarily Hermitian square matrix.

torch.linalg.svd() for a (slower) function that computes the more general SVD decomposition of matrices of any shape.

torch.linalg.qr() for another (much faster) decomposition that works on general matrices.

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 (tuple, optional) – output tuple of two tensors. Ignored if None. Default: None.

Returns

A named tuple (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.

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)
>>> L, Q = torch.linalg.eigh(A)
>>> L
tensor([0.3277, 2.9415], dtype=torch.float64)
>>> Q
tensor([[-0.0846+-0.0000j, -0.9964+0.0000j],
        [ 0.9170+0.3898j, -0.0779-0.0331j]], dtype=torch.complex128)
>>> torch.dist(Q @ torch.diag(L.cdouble()) @ Q.T.conj(), A)
tensor(6.1062e-16, dtype=torch.float64)

>>> A = torch.randn(3, 2, 2, dtype=torch.float64)
>>> A = A + A.transpose(-2, -1)  # creates a batch of symmetric matrices
>>> L, Q = torch.linalg.eigh(A)
>>> torch.dist(Q @ torch.diag_embed(L) @ Q.transpose(-2, -1).conj(), A)
tensor(1.5423e-15, dtype=torch.float64)