lucid.linalg.inv

lucid.linalg.inv(a: Tensor, /) → Tensor

The inv function computes the inverse of a square matrix.

Function Signature

def inv(a: Tensor) -> Tensor

Parameters

  • a (Tensor):

    The input tensor, which must be a square matrix (same number of rows and columns).

Returns

  • Tensor:

    A tensor representing the inverse of the input matrix.

Forward Calculation

The forward calculation for inv is based on the mathematical inverse of a matrix:

\[\mathbf{A}^{-1} \cdot \mathbf{A} = \mathbf{I}\]

where \(\mathbf{A}\) is the input matrix, and \(\mathbf{I}\) is the identity matrix.

Backward Gradient Calculation

For a tensor \(\mathbf{A}\) involved in the inv operation, the gradient of the inverse with respect to the input is computed as:

\[\frac{\partial \mathbf{A}^{-1}}{\partial \mathbf{A}} = -\mathbf{A}^{-1} \cdot \frac{\partial \mathbf{A}}{\partial \mathbf{A}} \cdot \mathbf{A}^{-1}\]

This result uses the property of the matrix inverse and chain rule to propagate gradients during backpropagation.

Raises

Attention

  • ValueError: If the input tensor is not a square matrix.

  • LinAlgError: If the matrix is singular and cannot be inverted.

Example

>>> import lucid
>>> a = lucid.Tensor([[1.0, 2.0], [3.0, 4.0]])
>>> inv_a = lucid.linalg.inv(a)
>>> print(inv_a)
Tensor([[-2.  1. ]
        [ 1.5 -0.5]])

Note

  • The input tensor must have two dimensions and be square, i.e., a.shape[0] == a.shape[1].

  • This function uses efficient numerical methods to compute the inverse but may raise an error if the matrix is poorly conditioned or singular.

Caution

Computing the inverse of a matrix can be numerically unstable for matrices that are close to singular.

Consider using other techniques such as LU decomposition or pseudo-inverses for better stability.