qr — linalg

qr

→Tensor

qr(x: Tensor, mode: str = 'reduced')

source

QR decomposition of a matrix.

Factorizes a matrix $A \in \mathbb{R}^{m \times n}$ as

A \,=\, Q\,R,

where $Q$ has orthonormal columns ( $Q^\top Q = I$ ) and $R$ is upper-triangular. Used for orthogonalizing a basis, solving least-squares ( $\min \|Ax - b\|_2$ ), and computing eigenvalues via QR iteration.

Parameters

xTensor

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

mode(reduced, complete, r)= "reduced"

"reduced" (default):

Q

is (m, k) and

R

is (k, n) with

k = \min(m, n)

. "complete":

Q

is (m, m) and

R

is (m, n). "r": return only

R

(Q is an empty tensor).

Returns

Tensor

Orthogonal factor.

Notes

Computed via Householder reflections (LAPACK geqrf). Cost is $O(2 m n^2 - \tfrac{2}{3} n^3)$ for $m \ge n$ .

The diagonal of $R$ may carry arbitrary signs (LAPACK convention) — the factorization is unique only up to a diagonal sign matrix. Backward routes $R$ through a Cholesky of $A^\top A$ (sign-robust) and $Q$ through the Stiefel-manifold tangent projection.

Examples

>>> import lucid
>>> from lucid.linalg import qr
>>> A = lucid.tensor([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> Q, R = qr(A)
>>> Q.T @ Q
Tensor([[1.0000, 0.0000],
        [0.0000, 1.0000]])