torch.linalg.eigvals

torch.linalg.eigvals(A, *, out=None) → Tensor

Computes the eigenvalues of a square matrix.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, the eigenvalues of a square matrix $A \in \mathbb{K}^{n \times n}$ are defined as the roots (counted with multiplicity) of the polynomial p of degree n given by

$$p(\lambda) = \operatorname{det}(A - \lambda \mathrm{I}_n)\mathrlap{\qquad \lambda \in \mathbb{C}}$$

where $\mathrm{I}_n$ is the n-dimensional identity matrix.

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.

Note

The eigenvalues of a real matrix may be complex, as the roots of a real polynomial may be complex.

The eigenvalues of a matrix are always well-defined, even when the matrix is not diagonalizable.

Note

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

See also

torch.linalg.eig() computes the full eigenvalue decomposition.

Parameters

A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions.

Keyword Arguments

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

Returns

A complex-valued tensor cointaining the eigenvalues even when A is real.

Examples:

>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a
tensor([[ 0.9828+0.3889j, -0.4617+0.3010j],
        [ 0.1662-0.7435j, -0.6139+0.0562j]], dtype=torch.complex128)
>>> w = torch.linalg.eigvals(a)
>>> w
tensor([ 1.1226+0.5738j, -0.7537-0.1286j], dtype=torch.complex128)