torch.linalg.inv

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

Computes the inverse of a square matrix if it exists. Throws a RuntimeError if the matrix is not invertible.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, for a matrix $A \in \mathbb{K}^{n \times n}$, its inverse matrix $A^{-1} \in \mathbb{K}^{n \times n}$ (if it exists) is defined as

$$A^{-1}A = AA^{-1} = \mathrm{I}_n$$

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

The inverse matrix exists if and only if $A$ is invertible. In this case, the inverse is unique.

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

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

Note

Consider using torch.linalg.solve() if possible for multiplying a matrix on the left by the inverse, as:

linalg.solve(A, B) == linalg.inv(A) @ B  # When B is a matrix

It is always preferred to use solve() when possible, as it is faster and more numerically stable than computing the inverse explicitly.

See also

torch.linalg.pinv() computes the pseudoinverse (Moore-Penrose inverse) of matrices of any shape.

torch.linalg.solve() computes A.inv() @ B with a numerically stable algorithm.

Parameters:

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

Keyword Arguments:

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

Raises:

RuntimeError – if the matrix A or any matrix in the batch of matrices A is not invertible.

Examples:

>>> A = torch.randn(4, 4)
>>> Ainv = torch.linalg.inv(A)
>>> torch.dist(A @ Ainv, torch.eye(4))
tensor(1.1921e-07)

>>> A = torch.randn(2, 3, 4, 4)  # Batch of matrices
>>> Ainv = torch.linalg.inv(A)
>>> torch.dist(A @ Ainv, torch.eye(4))
tensor(1.9073e-06)

>>> A = torch.randn(4, 4, dtype=torch.complex128)  # Complex matrix
>>> Ainv = torch.linalg.inv(A)
>>> torch.dist(A @ Ainv, torch.eye(4))
tensor(7.5107e-16, dtype=torch.float64)