matrix_norm

→Tensor

matrix_norm(x: Tensor, ord: int | float | str = 'fro', dim: tuple[int, int] = (-2, -1), keepdim: bool = False)

source edit

Compute a matrix norm.

Reduces the trailing two axes of an input to a scalar matrix norm. Supported orders:

"fro" — Frobenius norm $\|A\|_F = \big(\sum_{ij} |A_{ij}|^2\big)^{1/2}$ .
"nuc" — nuclear norm $\|A\|_* = \sum_i \sigma_i(A)$ (sum of singular values).
1 / -1 — max / min absolute column sum.
inf / -inf — max / min absolute row sum.
2 / -2 — largest / smallest singular value (spectral norm and its reciprocal).

Parameters

xTensor

Input of shape (*, m, n).

ord(int, float or str)= 'fro'

Norm order. Default "fro".

dimtuple of two ints= (-2, -1)

Axis pair identifying the matrix dimensions. Default (-2, -1).

keepdimbool= False

If True, reduced dims are retained with size 1.

Returns

Tensor

Matrix norm of each batch — real, also for a complex x.

Raises

DtypeMismatch

If x is an integer or bool tensor (a TypeError).

ValueError

If x has fewer than two dimensions, dim does not name two distinct axes, or ord is not one of the orders above.

NotImplementedError

If ord is "nuc", 2 or -2 and x is complex: those orders are read off the singular values, and svd has no complex kernel yet.

Notes

Spectral and nuclear norms require an SVD and so cost $O(\min(m,n)^2 \max(m,n))$ . Entry-wise norms reduce in a single pass.

The entry-wise orders ("fro", $\pm 1$ , $\pm\infty$ ) are taken through vector_norm, so a complex x is reduced through its magnitudes $|A_{ij}|$ exactly as a vector would be.

Examples

>>> import lucid
>>> from lucid.linalg import matrix_norm
>>> A = lucid.tensor([[3.0, 4.0], [0.0, 0.0]])
>>> matrix_norm(A, ord="fro")
tensor(5.)
>>> matrix_norm(lucid.tensor([[3 + 4j, 0j], [0j, 0j]]), ord="fro")
tensor(5.)

>>> import lucid >>> from lucid.linalg import matrix_norm >>> A = lucid.tensor([[3.0, 4.0], [0.0, 0.0]]) >>> matrix_norm(A, ord="fro") tensor(5.) >>> matrix_norm(lucid.tensor([[3 + 4j, 0j], [0j, 0j]]), ord="fro") tensor(5.)