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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 K\mathbb{K} be R\mathbb{R} or C\mathbb{C}, for a matrix AKn×nA \in \mathbb{K}^{n \times n}, its inverse matrix A1Kn×nA^{-1} \in \mathbb{K}^{n \times n} (if it exists) is defined as

A1A=AA1=InA^{-1}A = AA^{-1} = \mathrm{I}_n

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

The inverse matrix exists if and only if AA 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)

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