# torch.linalg.qr¶

torch.linalg.qr(A, mode='reduced', *, out=None)

Computes the QR decomposition of a matrix.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, the full QR decomposition of a matrix $A \in \mathbb{K}^{m \times n}$ is defined as

$A = QR\mathrlap{\qquad Q \in \mathbb{K}^{m \times m}, R \in \mathbb{K}^{m \times n}}$

where $Q$ is orthogonal in the real case and unitary in the complex case, and $R$ is upper triangular.

When m > n (tall matrix), as R is upper triangular, its last m - n rows are zero. In this case, we can drop the last m - n columns of Q to form the reduced QR decomposition:

$A = QR\mathrlap{\qquad Q \in \mathbb{K}^{m \times n}, R \in \mathbb{K}^{n \times n}}$

The reduced QR decomposition agrees with the full QR decomposition when n >= m (wide matrix).

Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A is a batch of matrices then the output has the same batch dimensions.

The parameter mode chooses between the full and reduced QR decomposition. If A has shape (*, m, n), denoting k = min(m, n)

• mode= ‘reduced’ (default): Returns (Q, R) of shapes (*, m, k), (*, k, n) respectively.

• mode= ‘complete’: Returns (Q, R) of shapes (*, m, m), (*, m, n) respectively.

• mode= ‘r’: Computes only the reduced R. Returns (Q, R) with Q empty and R of shape (*, k, n).

Differences with numpy.linalg.qr:

• mode= ‘raw’ is not implemented.

• Unlike numpy.linalg.qr, this function always returns a tuple of two tensors. When mode= ‘r’, the Q tensor is an empty tensor. This behavior may change in a future PyTorch release.

Note

The elements in the diagonal of R are not necessarily positive.

Note

mode= ‘r’ does not support backpropagation. Use mode= ‘reduced’ instead.

Warning

The QR decomposition is only unique up to the sign of the diagonal of R when the first k = min(m, n) columns of A are linearly independent. If this is not the case, different platforms, like NumPy, or inputs on different devices, may produce different valid decompositions.

Warning

Gradient computations are only supported if the first k = min(m, n) columns of every matrix in A are linearly independent. If this condition is not met, no error will be thrown, but the gradient produced will be incorrect. This is because the QR decomposition is not differentiable at these points.

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

• mode (str, optional) – one of ‘reduced’, ‘complete’, ‘r’. Controls the shape of the returned tensors. Default: ‘reduced’.

Keyword Arguments

out (tuple, optional) – output tuple of two tensors. Ignored if None. Default: None.

Returns

A named tuple (Q, R).

Examples:

>>> A = torch.tensor([[12., -51, 4], [6, 167, -68], [-4, 24, -41]])
>>> Q, R = torch.linalg.qr(A)
>>> Q
tensor([[-0.8571,  0.3943,  0.3314],
[-0.4286, -0.9029, -0.0343],
[ 0.2857, -0.1714,  0.9429]])
>>> R
tensor([[ -14.0000,  -21.0000,   14.0000],
[   0.0000, -175.0000,   70.0000],
[   0.0000,    0.0000,  -35.0000]])
>>> (Q @ R).round()
tensor([[  12.,  -51.,    4.],
[   6.,  167.,  -68.],
[  -4.,   24.,  -41.]])
>>> (Q.T @ Q).round()
tensor([[ 1.,  0.,  0.],
[ 0.,  1., -0.],
[ 0., -0.,  1.]])
>>> Q2, R2 = torch.linalg.qr(A, mode='r')
>>> Q2
tensor([])
>>> torch.equal(R, R2)
True
>>> A = torch.randn(3, 4, 5)
>>> Q, R = torch.linalg.qr(A, mode='complete')
>>> torch.dist(Q @ R, A)
tensor(1.6099e-06)
>>> torch.dist(Q.mT @ Q, torch.eye(4))
tensor(6.2158e-07)