torch.linalg.householder_product — PyTorch 2.7 documentation (original) (raw)

torch.linalg.householder_product(A, tau, *, out=None) → Tensor¶

Computes the first n columns of a product of Householder matrices.

Let K\mathbb{K} be R\mathbb{R} or C\mathbb{C}, and let A∈Km×nA \in \mathbb{K}^{m \times n} be a matrix with columns ai∈Kma_i \in \mathbb{K}^mfor i=1,…,mi=1,\ldots,m with m≥nm \geq n. Denote by bib_i the vector resulting from zeroing out the first i−1i-1 components of aia_i and setting to 1 the ii-th. For a vector τ∈Kk\tau \in \mathbb{K}^k with k≤nk \leq n, this function computes the first nn columns of the matrix

H1H2...HkwithHi=Im−τibibiHH_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i b_i b_i^{\text{H}}

where Im\mathrm{I}_m is the m-dimensional identity matrix and bHb^{\text{H}} is the conjugate transpose when bb is complex, and the transpose when bb is real-valued. The output matrix is the same size as the input matrix A.

See Representation of Orthogonal or Unitary Matrices for further details.

Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if the inputs are batches of matrices then the output has the same batch dimensions.

See also

torch.geqrf() can be used together with this function to form the Q from theqr() decomposition.

torch.ormqr() is a related function that computes the matrix multiplication of a product of Householder matrices with another matrix. However, that function is not supported by autograd.

Warning

Gradient computations are only well-defined if τi≠1∣∣ai∣∣2\tau_i \neq \frac{1}{||a_i||^2}. If this condition is not met, no error will be thrown, but the gradient produced may contain NaN.

Parameters

• A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions.
• tau (Tensor) – tensor of shape (*, k) where * is zero or more batch dimensions.

Keyword Arguments

out (Tensor, optional) – output tensor. Ignored if None. Default: None.

Raises

RuntimeError – if A doesn’t satisfy the requirement m >= n, or tau doesn’t satisfy the requirement n >= k.

Examples:

A = torch.randn(2, 2) h, tau = torch.geqrf(A) Q = torch.linalg.householder_product(h, tau) torch.dist(Q, torch.linalg.qr(A).Q) tensor(0.)

h = torch.randn(3, 2, 2, dtype=torch.complex128) tau = torch.randn(3, 1, dtype=torch.complex128) Q = torch.linalg.householder_product(h, tau) Q tensor([[[ 1.8034+0.4184j, 0.2588-1.0174j], [-0.6853+0.7953j, 2.0790+0.5620j]],

    [[ 1.4581+1.6989j, -1.5360+0.1193j],
    [ 1.3877-0.6691j,  1.3512+1.3024j]],

    [[ 1.4766+0.5783j,  0.0361+0.6587j],
    [ 0.6396+0.1612j,  1.3693+0.4481j]]], dtype=torch.complex128)