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

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

Letting K\mathbb{K} be R\mathbb{R} or C\mathbb{C}, for a matrix VKm×nV \in \mathbb{K}^{m \times n} with columns viKmv_i \in \mathbb{K}^m with mnm \geq n and a vector τKk\tau \in \mathbb{K}^k with knk \leq n, this function computes the first nn columns of the matrix

H1H2...HkwithHi=ImτiviviHH_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i v_i v_i^{\text{H}}

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


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.

  • 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.allclose(q, torch.linalg.qr(a)[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)


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