householder_product

→Tensor

householder_product(H: Tensor, tau: Tensor)

source

Reconstruct an orthogonal matrix from Householder reflectors.

Computes the implicit product

Q \,=\, H_1\,H_2\,\cdots\,H_k, \qquad H_i \,=\, I - \tau_i\, v_i\, v_i^\top,

where each $v_i$ is a Householder vector stored in the $i$ -th column of the packed input H and $\tau_i$ is its scalar factor. This is the standard way to materialise the $Q$ factor from a packed QR (geqrf) result.

Parameters

HTensor

Packed reflector matrix of shape (*, m, k) — columns contain the Householder vectors (typically the output of an unpacked geqrf).

tauTensor

Scalar factors of shape (*, k).

Returns

Tensor

Orthogonal matrix $Q$ of shape (*, m, k).

Notes

Backed by LAPACK orgqr. Cost is $O(m k^2)$ . Useful when a routine returns the packed Householder form (cheaper to store) but the explicit $Q$ is needed downstream.

Examples

>>> import lucid
>>> from lucid.linalg import qr, householder_product
>>> A = lucid.randn(4, 3)
>>> Q, R = qr(A, mode="reduced")
>>> # The same Q can be reconstructed from packed Householder reflectors
>>> # returned by lower-level geqrf-style factorisations.
>>> Q.shape
(4, 3)