torch.linalg.householder_product

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

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

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, for a matrix $V \in \mathbb{K}^{m \times n}$ with columns $v_i \in \mathbb{K}^m$ with $m \geq n$ and a vector $\tau \in \mathbb{K}^k$ with $k \leq n$, this function computes the first $n$ columns of the matrix

$$H_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i v_i v_i^{\text{H}}$$

where $\mathrm{I}_m$ is the m-dimensional identity matrix and $v^{\text{H}}$ is the conjugate transpose when $v$ is complex, and the transpose when $v$ is real-valued.

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.

Note

This function only uses the values strictly below the main diagonal of A. The other values are ignored.

See also

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

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)