inv

→Tensor

inv(x: Tensor)

source edit

Implementing kernel

Compute the multiplicative inverse of a square matrix.

Returns the unique matrix $A^{-1}$ such that

A A^{-1} = A^{-1} A = I

Inversion is performed via LU decomposition with partial pivoting. When the goal is to apply $A^{-1}$ to a known right-hand side, prefer solve — explicit inversion is both slower and less numerically stable than back-substitution.

Parameters

xTensor

Square matrix of shape (*, n, n) (batch dims allowed). Must be non-singular; raises a runtime error on singular input.

Returns

Tensor

Inverse matrix of shape (*, n, n) with the same dtype as x.

Notes

Internally computes $PA = LU$ and then solves LU X = P for X via two triangular sweeps. Cost is $O(n^3)$ per matrix. For ill-conditioned A consider pinv (SVD-based) for a more robust pseudo-inverse.

A matrix with a zero dimension is returned unchanged rather than handed to the factorisation. The 0x0 matrix is its own inverse — it is the identity on the zero-dimensional space — so nothing is lost, and LAPACK's behaviour at n = 0 is not uniform across the implementations this runs against: the same call is harmless on one Accelerate build and a segmentation fault on another. Guarding here makes the answer depend on the mathematics rather than on which machine is asking.

Examples

>>> import lucid
>>> from lucid.linalg import inv
>>> A = lucid.tensor([[4.0, 7.0], [2.0, 6.0]])
>>> inv(A)
tensor([[0.6, -0.7], [-0.2, 0.4]])

inv