matrix_rank

→Tensor

matrix_rank(A: Tensor, tol: float | None = None, hermitian: bool = False)

source

Compute the numerical rank of a matrix.

The numerical rank is the number of singular values of $A$ that exceed a tolerance:

\mathrm{rank}(A) \,=\, |\{\,i : \sigma_i > \tau\,\}|.

Parameters

ATensor

Input matrix of shape (*, m, n).

tolfloat or None= None

Threshold below which singular values are treated as zero. None (default) uses

\max(m, n) \cdot \varepsilon \cdot \sigma_{\max}

with

\varepsilon

the float32 machine epsilon (

\approx 1.19 \times 10^{-7}

hermitianbool= False

If True exploit Hermitian structure for a cheaper eigen-based computation. Currently unused (SVD path is always taken).

Returns

Tensor

Integer scalar (or batched scalars) holding the rank.

Notes

Computed via svd so cost is $O(\min(m,n)^2 \cdot \max(m,n))$ . Choosing the tolerance is application-specific — consider scaling by $\|A\|$ when the singular values span many orders of magnitude.

Examples

>>> import lucid
>>> from lucid.linalg import matrix_rank
>>> matrix_rank(lucid.tensor([[1.0, 2.0], [2.0, 4.0]]))
Tensor(1)