torch.linalg.matrix_exp

torch.linalg.matrix_exp(A) → Tensor

Computes the matrix exponential of a square matrix.

Letting \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), this function computes the matrix exponential of \(A \in \mathbb{K}^{n \times n}\), which is defined as

\[\mathrm{matrix\_exp}(A) = \sum_{k=0}^\infty \frac{1}{k!}A^k \in \mathbb{K}^{n \times n}. \]

If the matrix \(A\) has eigenvalues \(\lambda_i \in \mathbb{C}\), the matrix \(\mathrm{matrix\_exp}(A)\) has eigenvalues \(e^{\lambda_i} \in \mathbb{C}\).

Supports input of bfloat16, 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.

Parameters:

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

Example:

>>> A = torch.empty(2, 2, 2)
>>> A[0, :, :] = torch.eye(2, 2)
>>> A[1, :, :] = 2 * torch.eye(2, 2)
>>> A
tensor([[[1., 0.],
         [0., 1.]],

        [[2., 0.],
         [0., 2.]]])
>>> torch.linalg.matrix_exp(A)
tensor([[[2.7183, 0.0000],
         [0.0000, 2.7183]],

         [[7.3891, 0.0000],
          [0.0000, 7.3891]]])

>>> import math
>>> A = torch.tensor([[0, math.pi/3], [-math.pi/3, 0]]) # A is skew-symmetric
>>> torch.linalg.matrix_exp(A) # matrix_exp(A) = [[cos(pi/3), sin(pi/3)], [-sin(pi/3), cos(pi/3)]]
tensor([[ 0.5000,  0.8660],
        [-0.8660,  0.5000]])