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 VKm×nV \in \mathbb{K}^{m \times n} be a matrix with columns viKmv_i \in \mathbb{K}^m for i=1,,mi=1,\ldots,m with mnm \geq n. Denote by wiw_i the vector resulting from zeroing out the first i1i-1 components of viv_i and setting to 1 the ii-th. For a vector τKk\tau \in \mathbb{K}^k with knk \leq n, this function computes the first nn columns of the matrix

H1H2...HkwithHi=ImτiwiwiHH_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i w_i w_i^{\text{H}}

where Im\mathrm{I}_m is the m-dimensional identity matrix and wHw^{\text{H}} is the conjugate transpose when ww is complex, and the transpose when ww 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 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.


Gradient computations are only well-defined if taui1vi2tau_i \neq \frac{1}{||v_i||^2}. If this condition is not met, no error will be thrown, but the gradient produced may contain NaN.

  • 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.


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


>>> 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)

>>> 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)


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