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

Source: R/linalg.R
linalg_eig.Rd

Letting K be R or C, the eigenvalue decomposition of a square matrix A K^n 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.

Value

A list (eigenvalues, eigenvectors) which corresponds to 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.

Details

A=Vdiag⁡(Λ)V−1V∈Cn×n,Λ∈Cn A = V \operatorname{diag}(\Lambda) V^{-1}\mathrlap{\qquad V \in \mathbb{C}^{n \times n}, \Lambda \in \mathbb{C}^n}

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.

Note

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

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 V diag()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 _i through the computation of 1_i j _i - _j.

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(), linalg_cholesky_ex(), linalg_det(), linalg_eigh(), linalg_eigvals(), linalg_eigvalsh(), linalg_householder_product(), linalg_inv(), linalg_inv_ex(), 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_solve_triangular(), linalg_svd(), linalg_svdvals(), linalg_tensorinv(), linalg_tensorsolve(), linalg_vector_norm()

Examples

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