linalg_eig {torch}

Computes the eigenvalue decomposition of a square matrix if it exists.

Description

Letting \mathbb{K} be \mathbb{R} or \mathbb{C}, the eigenvalue decomposition of a square matrix A \in \mathbb{K}^{n \times n} (if it exists) is defined as

Usage

linalg_eig(A)

Arguments

A

(Tensor): tensor of shape ⁠(*, n, n)⁠ where * is zero or more batch dimensions consisting of diagonalizable matrices.

Details

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This decomposition exists if and only if A is diagonalizable_. This is the case when all its eigenvalues are different. 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.

Value

A list ⁠(eigenvalues, eigenvectors)⁠ which corresponds to \Lambda and V above. eigenvalues and eigenvectors will always be complex-valued, even when A is real. The eigenvectors will be given by the columns of eigenvectors.

Warning

• This function assumes that A is diagonalizable_ (for example, when all the eigenvalues are different). If it is not diagonalizable, the returned eigenvalues will be correct but A \neq V \operatorname{diag}(\Lambda)V^{-1}.

• The eigenvectors of a 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 a non-zero number produces another set of valid eigenvectors of the matrix. In this implmentation, the returned eigenvectors are normalized to have norm 1 and largest real component.

• Gradients computed using V will only be finite when A does not have repeated 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}.

Note

The eigenvalues and eigenvectors of a real matrix may be complex.

See Also

• linalg_eigvals() computes only the eigenvalues. Unlike linalg_eig(), the gradients of linalg_eigvals() are always numerically stable.

• linalg_eigh() for a (faster) function that computes the eigenvalue decomposition for Hermitian and symmetric matrices.

• linalg_svd() for a function that computes another type of spectral decomposition that works on matrices of any shape.

• linalg_qr() for another (much faster) decomposition that works on matrices of any shape.

Other linalg: linalg_cholesky_ex(), linalg_cholesky(), linalg_det(), linalg_eigh(), linalg_eigvalsh(), linalg_eigvals(), 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(), linalg_svdvals(), linalg_svd(), linalg_tensorinv(), linalg_tensorsolve(), linalg_vector_norm()

Examples

if (torch_is_installed()) {
a <- torch_randn(2, 2)
wv <- linalg_eig(a)
}