# torch.linalg¶

Common linear algebra operations.

This module is in BETA. New functions are still being added, and some functions may change in future PyTorch releases. See the documentation of each function for details.

## Functions¶

torch.linalg.cholesky(input, *, out=None) → Tensor

Computes the Cholesky decomposition of a Hermitian (or symmetric for real-valued matrices) positive-definite matrix or the Cholesky decompositions for a batch of such matrices. Each decomposition has the form:

$\text{input} = LL^H$

where $L$ is a lower-triangular matrix and $L^H$ is the conjugate transpose of $L$ , which is just a transpose for the case of real-valued input matrices. In code it translates to input = L @ L.t() if input is real-valued and input = L @ L.conj().t() if input is complex-valued. The batch of $L$ matrices is returned.

Supports real-valued and complex-valued inputs.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

LAPACK’s potrf is used for CPU inputs, and MAGMA’s potrf is used for CUDA inputs.

Note

If input is not a Hermitian positive-definite matrix, or if it’s a batch of matrices and one or more of them is not a Hermitian positive-definite matrix, then a RuntimeError will be thrown. If input is a batch of matrices, then the error message will include the batch index of the first matrix that is not Hermitian positive-definite.

Parameters

input (Tensor) – the input tensor of size $(*, n, n)$ consisting of Hermitian positive-definite $n \times n$ matrices, where $*$ is zero or more batch dimensions.

Keyword Arguments

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

Examples:

>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = torch.mm(a, a.t().conj())  # creates a Hermitian positive-definite matrix
>>> l = torch.linalg.cholesky(a)
>>> a
tensor([[2.5266+0.0000j, 1.9586-2.0626j],
[1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)
>>> l
tensor([[1.5895+0.0000j, 0.0000+0.0000j],
[1.2322+1.2976j, 2.4928+0.0000j]], dtype=torch.complex128)
>>> torch.mm(l, l.t().conj())
tensor([[2.5266+0.0000j, 1.9586-2.0626j],
[1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)

>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = torch.matmul(a, a.transpose(-2, -1))  # creates a symmetric positive-definite matrix
>>> l = torch.linalg.cholesky(a)
>>> a
tensor([[[ 1.1629,  2.0237],
[ 2.0237,  6.6593]],

[[ 0.4187,  0.1830],
[ 0.1830,  0.1018]],

[[ 1.9348, -2.5744],
[-2.5744,  4.6386]]], dtype=torch.float64)
>>> l
tensor([[[ 1.0784,  0.0000],
[ 1.8766,  1.7713]],

[[ 0.6471,  0.0000],
[ 0.2829,  0.1477]],

[[ 1.3910,  0.0000],
[-1.8509,  1.1014]]], dtype=torch.float64)
>>> torch.allclose(torch.matmul(l, l.transpose(-2, -1)), a)
True

torch.linalg.cond(input, p=None, *, out=None) → Tensor

Computes the condition number of a matrix input, or of each matrix in a batched input, using the matrix norm defined by p.

For norms {‘fro’, ‘nuc’, inf, -inf, 1, -1} this is defined as the matrix norm of input times the matrix norm of the inverse of input computed using torch.linalg.norm(). While for norms {None, 2, -2} this is defined as the ratio between the largest and smallest singular values computed using torch.linalg.svd().

This function supports float, double, cfloat and cdouble dtypes.

Note

When given inputs on a CUDA device, this function may synchronize that device with the CPU depending on which norm p is used.

Note

For norms {None, 2, -2}, input may be a non-square matrix or batch of non-square matrices. For other norms, however, input must be a square matrix or a batch of square matrices, and if this requirement is not satisfied a RuntimeError will be thrown.

Note

For norms {‘fro’, ‘nuc’, inf, -inf, 1, -1} if input is a non-invertible matrix then a tensor containing infinity will be returned. If input is a batch of matrices and one or more of them is not invertible then a RuntimeError will be thrown.

Parameters
• input (Tensor) – the input matrix of size (m, n) or the batch of matrices of size (*, m, n) where * is one or more batch dimensions.

• p (int, float, inf, -inf, 'fro', 'nuc', optional) –

the type of the matrix norm to use in the computations. inf refers to float('inf'), numpy’s inf object, or any equivalent object. The following norms can be used:

p

norm for matrices

None

ratio of the largest singular value to the smallest singular value

’fro’

Frobenius norm

’nuc’

nuclear norm

inf

max(sum(abs(x), dim=1))

-inf

min(sum(abs(x), dim=1))

1

max(sum(abs(x), dim=0))

-1

min(sum(abs(x), dim=0))

2

ratio of the largest singular value to the smallest singular value

-2

ratio of the smallest singular value to the largest singular value

Default: None

Keyword Arguments

out (Tensor, optional) – tensor to write the output to. Default is None.

Returns

The condition number of input. The output dtype is always real valued even for complex inputs (e.g. float if input is cfloat).

Examples:

>>> a = torch.randn(3, 4, 4, dtype=torch.complex64)
>>> torch.linalg.cond(a)
>>> a = torch.tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]])
>>> torch.linalg.cond(a)
tensor([1.4142])
>>> torch.linalg.cond(a, 'fro')
tensor(3.1623)
>>> torch.linalg.cond(a, 'nuc')
tensor(9.2426)
>>> torch.linalg.cond(a, float('inf'))
tensor(2.)
>>> torch.linalg.cond(a, float('-inf'))
tensor(1.)
>>> torch.linalg.cond(a, 1)
tensor(2.)
>>> torch.linalg.cond(a, -1)
tensor(1.)
>>> torch.linalg.cond(a, 2)
tensor([1.4142])
>>> torch.linalg.cond(a, -2)
tensor([0.7071])

>>> a = torch.randn(2, 3, 3)
>>> a
tensor([[[-0.9204,  1.1140,  1.2055],
[ 0.3988, -0.2395, -0.7441],
[-0.5160,  0.3115,  0.2619]],

[[-2.2128,  0.9241,  2.1492],
[-1.1277,  2.7604, -0.8760],
[ 1.2159,  0.5960,  0.0498]]])
>>> torch.linalg.cond(a)
tensor([[9.5917],
[3.2538]])

>>> a = torch.randn(2, 3, 3, dtype=torch.complex64)
>>> a
tensor([[[-0.4671-0.2137j, -0.1334-0.9508j,  0.6252+0.1759j],
[-0.3486-0.2991j, -0.1317+0.1252j,  0.3025-0.1604j],
[-0.5634+0.8582j,  0.1118-0.4677j, -0.1121+0.7574j]],

[[ 0.3964+0.2533j,  0.9385-0.6417j, -0.0283-0.8673j],
[ 0.2635+0.2323j, -0.8929-1.1269j,  0.3332+0.0733j],
[ 0.1151+0.1644j, -1.1163+0.3471j, -0.5870+0.1629j]]])
>>> torch.linalg.cond(a)
tensor([[4.6245],
[4.5671]])
>>> torch.linalg.cond(a, 1)
tensor([9.2589, 9.3486])

torch.linalg.det(input) → Tensor

Computes the determinant of a square matrix input, or of each square matrix in a batched input.

This function supports float, double, cfloat and cdouble dtypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The determinant is computed using LU factorization. LAPACK’s getrf is used for CPU inputs, and MAGMA’s getrf is used for CUDA inputs.

Note

Backward through det internally uses torch.linalg.svd() when input is not invertible. In this case, double backward through det will be unstable when input doesn’t have distinct singular values. See torch.linalg.svd() for more details.

Parameters

input (Tensor) – the input matrix of size (n, n) or the batch of matrices of size (*, n, n) where * is one or more batch dimensions.

Example:

>>> a = torch.randn(3, 3)
>>> a
tensor([[ 0.9478,  0.9158, -1.1295],
[ 0.9701,  0.7346, -1.8044],
[-0.2337,  0.0557,  0.6929]])
>>> torch.linalg.det(a)
tensor(0.0934)

>>> a = torch.randn(3, 2, 2)
>>> a
tensor([[[ 0.9254, -0.6213],
[-0.5787,  1.6843]],

[[ 0.3242, -0.9665],
[ 0.4539, -0.0887]],

[[ 1.1336, -0.4025],
[-0.7089,  0.9032]]])
>>> torch.linalg.det(a)
tensor([1.1990, 0.4099, 0.7386])

torch.linalg.slogdet(input, *, out=None) -> (Tensor, Tensor)

Calculates the sign and natural logarithm of the absolute value of a square matrix’s determinant, or of the absolute values of the determinants of a batch of square matrices input. The determinant can be computed with sign * exp(logabsdet).

Supports input of float, double, cfloat and cdouble datatypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The determinant is computed using LU factorization. LAPACK’s getrf is used for CPU inputs, and MAGMA’s getrf is used for CUDA inputs.

Note

For matrices that have zero determinant, this returns (0, -inf). If input is batched then the entries in the result tensors corresponding to matrices with the zero determinant have sign 0 and the natural logarithm of the absolute value of the determinant -inf.

Parameters

input (Tensor) – the input matrix of size $(n, n)$ or the batch of matrices of size $(*, n, n)$ where $*$ is one or more batch dimensions.

Keyword Arguments

out (tuple, optional) – tuple of two tensors to write the output to.

Returns

A namedtuple (sign, logabsdet) containing the sign of the determinant and the natural logarithm of the absolute value of determinant, respectively.

Example:

>>> A = torch.randn(3, 3)
>>> A
tensor([[ 0.0032, -0.2239, -1.1219],
[-0.6690,  0.1161,  0.4053],
[-1.6218, -0.9273, -0.0082]])
>>> torch.linalg.det(A)
tensor(-0.7576)
>>> torch.linalg.logdet(A)
tensor(nan)
>>> torch.linalg.slogdet(A)
torch.return_types.linalg_slogdet(sign=tensor(-1.), logabsdet=tensor(-0.2776))

torch.linalg.eigh(input, UPLO='L', *, out=None) -> (Tensor, Tensor)

Computes the eigenvalues and eigenvectors of a complex Hermitian (or real symmetric) matrix input, or of each such matrix in a batched input.

For a single matrix input, the tensor of eigenvalues w and the tensor of eigenvectors V decompose the input such that input = V diag(w) Vᴴ, where Vᴴ is the transpose of V for real-valued input, or the conjugate transpose of V for complex-valued input.

Since the matrix or matrices in input are assumed to be Hermitian, the imaginary part of their diagonals is always treated as zero. When UPLO is “L”, its default value, only the lower triangular part of each matrix is used in the computation. When UPLO is “U” only the upper triangular part of each matrix is used.

Supports input of float, double, cfloat and cdouble dtypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The eigenvalues/eigenvectors are computed using LAPACK’s syevd and heevd routines for CPU inputs, and MAGMA’s syevd and heevd routines for CUDA inputs.

Note

The eigenvalues of real symmetric or complex Hermitian matrices are always real.

Note

The eigenvectors of matrices are not unique, so any eigenvector multiplied by a constant remains a valid eigenvector. This function may compute different eigenvector representations on different device types. Usually the difference is only in the sign of the eigenvector.

Note

See torch.linalg.eigvalsh() for a related function that computes only eigenvalues. However, that function is not differentiable.

Parameters
• input (Tensor) – the Hermitian n times n matrix or the batch of such matrices of size (*, n, n) where * is one or more batch dimensions.

• UPLO ('L', 'U', optional) – controls whether to use the upper-triangular or the lower-triangular part of input in the computations. Default is 'L'.

Keyword Arguments

out (tuple, optional) – tuple of two tensors to write the output to. Default is None.

Returns

A namedtuple (eigenvalues, eigenvectors) containing

• eigenvalues (Tensor): Shape (*, m).

The eigenvalues in ascending order.

• eigenvectors (Tensor): Shape (*, m, m).

The orthonormal eigenvectors of the input.

Return type

(Tensor, Tensor)

Examples:

>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = a + a.t().conj()  # creates a Hermitian matrix
>>> a
tensor([[2.9228+0.0000j, 0.2029-0.0862j],
[0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)
>>> w, v = torch.linalg.eigh(a)
>>> w
tensor([0.3277, 2.9415], dtype=torch.float64)
>>> v
tensor([[-0.0846+-0.0000j, -0.9964+0.0000j],
[ 0.9170+0.3898j, -0.0779-0.0331j]], dtype=torch.complex128)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.to(v.dtype).diag_embed(), v.t().conj())), a)
True

>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = a + a.transpose(-2, -1)  # creates a symmetric matrix
>>> w, v = torch.linalg.eigh(a)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.diag_embed(), v.transpose(-2, -1))), a)
True

torch.linalg.eigvalsh(input, UPLO='L', *, out=None) → Tensor

Computes the eigenvalues of a complex Hermitian (or real symmetric) matrix input, or of each such matrix in a batched input. The eigenvalues are returned in ascending order.

Since the matrix or matrices in input are assumed to be Hermitian, the imaginary part of their diagonals is always treated as zero. When UPLO is “L”, its default value, only the lower triangular part of each matrix is used in the computation. When UPLO is “U” only the upper triangular part of each matrix is used.

Supports input of float, double, cfloat and cdouble dtypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The eigenvalues are computed using LAPACK’s syevd and heevd routines for CPU inputs, and MAGMA’s syevd and heevd routines for CUDA inputs.

Note

The eigenvalues of real symmetric or complex Hermitian matrices are always real.

Note

This function doesn’t support backpropagation, please use torch.linalg.eigh() instead, which also computes the eigenvectors.

Note

See torch.linalg.eigh() for a related function that computes both eigenvalues and eigenvectors.

Parameters
• input (Tensor) – the Hermitian n times n matrix or the batch of such matrices of size (*, n, n) where * is one or more batch dimensions.

• UPLO ('L', 'U', optional) – controls whether to use the upper-triangular or the lower-triangular part of input in the computations. Default is 'L'.

Keyword Arguments

out (Tensor, optional) – tensor to write the output to. Default is None.

Examples:

>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = a + a.t().conj()  # creates a Hermitian matrix
>>> a
tensor([[2.9228+0.0000j, 0.2029-0.0862j],
[0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)
>>> w = torch.linalg.eigvalsh(a)
>>> w
tensor([0.3277, 2.9415], dtype=torch.float64)

>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = a + a.transpose(-2, -1)  # creates a symmetric matrix
>>> a
tensor([[[ 2.8050, -0.3850],
[-0.3850,  3.2376]],

[[-1.0307, -2.7457],
[-2.7457, -1.7517]],

[[ 1.7166,  2.2207],
[ 2.2207, -2.0898]]], dtype=torch.float64)
>>> w = torch.linalg.eigvalsh(a)
>>> w
tensor([[ 2.5797,  3.4629],
[-4.1605,  1.3780],
[-3.1113,  2.7381]], dtype=torch.float64)

torch.linalg.matrix_rank(input, tol=None, hermitian=False, *, out=None) → Tensor

Computes the numerical rank of a matrix input, or of each matrix in a batched input.

The matrix rank is computed as the number of singular values (or absolute eigenvalues when hermitian is True) that are greater than the specified tol threshold.

If tol is not specified, tol is set to S.max(dim=-1)*max(input.shape[-2:])*eps, where S is the singular values (or absolute eigenvalues when hermitian is True), and eps is the epsilon value for the datatype of input. The epsilon value can be obtained using the eps attribute of torch.finfo.

Supports input of float, double, cfloat and cdouble dtypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The matrix rank is computed using singular value decomposition (see torch.linalg.svd()) by default. If hermitian is True, then input is assumed to be Hermitian (symmetric if real-valued), and the computation is done by obtaining the eigenvalues (see torch.linalg.eigvalsh()).

Parameters
• input (Tensor) – the input matrix of size (m, n) or the batch of matrices of size (*, m, n) where * is one or more batch dimensions.

• tol (float, optional) – the tolerance value. Default is None

• hermitian (bool, optional) – indicates whether input is Hermitian. Default is False.

Keyword Arguments

out (Tensor, optional) – tensor to write the output to. Default is None.

Examples:

>>> a = torch.eye(10)
>>> torch.linalg.matrix_rank(a)
tensor(10)
>>> b = torch.eye(10)
>>> b[0, 0] = 0
>>> torch.linalg.matrix_rank(b)
tensor(9)

>>> a = torch.randn(4, 3, 2)
>>> torch.linalg.matrix_rank(a)
tensor([2, 2, 2, 2])

>>> a = torch.randn(2, 4, 2, 3)
>>> torch.linalg.matrix_rank(a)
tensor([[2, 2, 2, 2],
[2, 2, 2, 2]])

>>> a = torch.randn(2, 4, 3, 3, dtype=torch.complex64)
>>> torch.linalg.matrix_rank(a)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(a, hermitian=True)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(a, tol=1.0)
tensor([[3, 2, 2, 2],
[1, 2, 1, 2]])
>>> torch.linalg.matrix_rank(a, tol=1.0, hermitian=True)
tensor([[2, 2, 2, 1],
[1, 2, 2, 2]])

torch.linalg.norm(input, ord=None, dim=None, keepdim=False, *, out=None, dtype=None) → Tensor

Returns the matrix norm or vector norm of a given tensor.

This function can calculate one of eight different types of matrix norms, or one of an infinite number of vector norms, depending on both the number of reduction dimensions and the value of the ord parameter.

Parameters
• input (Tensor) – The input tensor. If dim is None, x must be 1-D or 2-D, unless ord is None. If both dim and ord are None, the 2-norm of the input flattened to 1-D will be returned. Its data type must be either a floating point or complex type. For complex inputs, the norm is calculated on of the absolute values of each element. If the input is complex and neither dtype nor out is specified, the result’s data type will be the corresponding floating point type (e.g. float if input is complexfloat).

• ord (int, float, inf, -inf, 'fro', 'nuc', optional) –

The order of norm. inf refers to float('inf'), numpy’s inf object, or any equivalent object. The following norms can be calculated:

ord

norm for matrices

norm for vectors

None

Frobenius norm

2-norm

’fro’

Frobenius norm

– not supported –

‘nuc’

nuclear norm

– not supported –

inf

max(sum(abs(x), dim=1))

max(abs(x))

-inf

min(sum(abs(x), dim=1))

min(abs(x))

0

– not supported –

sum(x != 0)

1

max(sum(abs(x), dim=0))

as below

-1

min(sum(abs(x), dim=0))

as below

2

2-norm (largest sing. value)

as below

-2

smallest singular value

as below

other

– not supported –

sum(abs(x)**ord)**(1./ord)

Default: None

• dim (int, 2-tuple of python:ints, 2-list of python:ints, optional) – If dim is an int, vector norm will be calculated over the specified dimension. If dim is a 2-tuple of ints, matrix norm will be calculated over the specified dimensions. If dim is None, matrix norm will be calculated when the input tensor has two dimensions, and vector norm will be calculated when the input tensor has one dimension. Default: None

• keepdim (bool, optional) – If set to True, the reduced dimensions are retained in the result as dimensions with size one. Default: False

Keyword Arguments
• out (Tensor, optional) – The output tensor. Ignored if None. Default: None

• dtype (torch.dtype, optional) – If specified, the input tensor is cast to dtype before performing the operation, and the returned tensor’s type will be dtype. If this argument is used in conjunction with the out argument, the output tensor’s type must match this argument or a RuntimeError will be raised. Default: None

Examples:

>>> import torch
>>> from torch import linalg as LA
>>> a = torch.arange(9, dtype=torch.float) - 4
>>> a
tensor([-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])
>>> b = a.reshape((3, 3))
>>> b
tensor([[-4., -3., -2.],
[-1.,  0.,  1.],
[ 2.,  3.,  4.]])

>>> LA.norm(a)
tensor(7.7460)
>>> LA.norm(b)
tensor(7.7460)
>>> LA.norm(b, 'fro')
tensor(7.7460)
>>> LA.norm(a, float('inf'))
tensor(4.)
>>> LA.norm(b, float('inf'))
tensor(9.)
>>> LA.norm(a, -float('inf'))
tensor(0.)
>>> LA.norm(b, -float('inf'))
tensor(2.)

>>> LA.norm(a, 1)
tensor(20.)
>>> LA.norm(b, 1)
tensor(7.)
>>> LA.norm(a, -1)
tensor(0.)
>>> LA.norm(b, -1)
tensor(6.)
>>> LA.norm(a, 2)
tensor(7.7460)
>>> LA.norm(b, 2)
tensor(7.3485)

>>> LA.norm(a, -2)
tensor(0.)
>>> LA.norm(b.double(), -2)
tensor(1.8570e-16, dtype=torch.float64)
>>> LA.norm(a, 3)
tensor(5.8480)
>>> LA.norm(a, -3)
tensor(0.)


Using the dim argument to compute vector norms:

>>> c = torch.tensor([[1., 2., 3.],
...                   [-1, 1, 4]])
>>> LA.norm(c, dim=0)
tensor([1.4142, 2.2361, 5.0000])
>>> LA.norm(c, dim=1)
tensor([3.7417, 4.2426])
>>> LA.norm(c, ord=1, dim=1)
tensor([6., 6.])


Using the dim argument to compute matrix norms:

>>> m = torch.arange(8, dtype=torch.float).reshape(2, 2, 2)
>>> LA.norm(m, dim=(1,2))
tensor([ 3.7417, 11.2250])
>>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
(tensor(3.7417), tensor(11.2250))

torch.linalg.pinv(input, rcond=1e-15, hermitian=False, *, out=None) → Tensor

Computes the pseudo-inverse (also known as the Moore-Penrose inverse) of a matrix input, or of each matrix in a batched input.

The singular values (or the absolute values of the eigenvalues when hermitian is True) that are below the specified rcond threshold are treated as zero and discarded in the computation.

Supports input of float, double, cfloat and cdouble datatypes.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The pseudo-inverse is computed using singular value decomposition (see torch.linalg.svd()) by default. If hermitian is True, then input is assumed to be Hermitian (symmetric if real-valued), and the computation of the pseudo-inverse is done by obtaining the eigenvalues and eigenvectors (see torch.linalg.eigh()).

Note

If singular value decomposition or eigenvalue decomposition algorithms do not converge then a RuntimeError will be thrown.

Parameters
• input (Tensor) – the input matrix of size (m, n) or the batch of matrices of size (*, m, n) where * is one or more batch dimensions.

• rcond (float, Tensor, optional) – the tolerance value to determine the cutoff for small singular values. Must be broadcastable to the singular values of input as returned by torch.svd(). Default is 1e-15.

• hermitian (bool, optional) – indicates whether input is Hermitian. Default is False.

Keyword Arguments

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

Examples:

>>> input = torch.randn(3, 5)
>>> input
tensor([[ 0.5495,  0.0979, -1.4092, -0.1128,  0.4132],
[-1.1143, -0.3662,  0.3042,  1.6374, -0.9294],
[-0.3269, -0.5745, -0.0382, -0.5922, -0.6759]])
>>> torch.linalg.pinv(input)
tensor([[ 0.0600, -0.1933, -0.2090],
[-0.0903, -0.0817, -0.4752],
[-0.7124, -0.1631, -0.2272],
[ 0.1356,  0.3933, -0.5023],
[-0.0308, -0.1725, -0.5216]])

Batched linalg.pinv example
>>> a = torch.randn(2, 6, 3)
>>> b = torch.linalg.pinv(a)
>>> torch.matmul(b, a)
tensor([[[ 1.0000e+00,  1.6391e-07, -1.1548e-07],
[ 8.3121e-08,  1.0000e+00, -2.7567e-07],
[ 3.5390e-08,  1.4901e-08,  1.0000e+00]],

[[ 1.0000e+00, -8.9407e-08,  2.9802e-08],
[-2.2352e-07,  1.0000e+00,  1.1921e-07],
[ 0.0000e+00,  8.9407e-08,  1.0000e+00]]])

Hermitian input example
>>> a = torch.randn(3, 3, dtype=torch.complex64)
>>> a = a + a.t().conj()  # creates a Hermitian matrix
>>> b = torch.linalg.pinv(a, hermitian=True)
>>> torch.matmul(b, a)
tensor([[ 1.0000e+00+0.0000e+00j, -1.1921e-07-2.3842e-07j,
5.9605e-08-2.3842e-07j],
[ 5.9605e-08+2.3842e-07j,  1.0000e+00+2.3842e-07j,
-4.7684e-07+1.1921e-07j],
[-1.1921e-07+0.0000e+00j, -2.3842e-07-2.9802e-07j,
1.0000e+00-1.7897e-07j]])

Non-default rcond example
>>> rcond = 0.5
>>> a = torch.randn(3, 3)
>>> torch.linalg.pinv(a)
tensor([[ 0.2971, -0.4280, -2.0111],
[-0.0090,  0.6426, -0.1116],
[-0.7832, -0.2465,  1.0994]])
>>> torch.linalg.pinv(a, rcond)
tensor([[-0.2672, -0.2351, -0.0539],
[-0.0211,  0.6467, -0.0698],
[-0.4400, -0.3638, -0.0910]])

Matrix-wise rcond example
>>> a = torch.randn(5, 6, 2, 3, 3)
>>> rcond = torch.rand(2)  # different rcond values for each matrix in a[:, :, 0] and a[:, :, 1]
>>> torch.linalg.pinv(a, rcond)
>>> rcond = torch.randn(5, 6, 2) # different rcond value for each matrix in 'a'
>>> torch.linalg.pinv(a, rcond)

torch.linalg.svd(input, full_matrices=True, compute_uv=True, *, out=None) -> (Tensor, Tensor, Tensor)

Computes the singular value decomposition of either a matrix or batch of matrices input.” The singular value decomposition is represented as a namedtuple (U, S, Vh), such that $input = U \mathbin{@} diag(S) \times Vh$ . If input is a batch of tensors, then U, S, and Vh are also batched with the same batch dimensions as input.

If full_matrices is False (default), the method returns the reduced singular value decomposition i.e., if the last two dimensions of input are m and n, then the returned U and V matrices will contain only $min(n, m)$ orthonormal columns.

If compute_uv is False, the returned U and Vh will be empy tensors with no elements and the same device as input. The full_matrices argument has no effect when compute_uv is False.

The dtypes of U and V are the same as input’s. S will always be real-valued, even if input is complex.

Note

Unlike NumPy’s linalg.svd, this always returns a namedtuple of three tensors, even when compute_uv=False. This behavior may change in a future PyTorch release.

Note

The singular values are returned in descending order. If input is a batch of matrices, then the singular values of each matrix in the batch is returned in descending order.

Note

The implementation of SVD on CPU uses the LAPACK routine ?gesdd (a divide-and-conquer algorithm) instead of ?gesvd for speed. Analogously, the SVD on GPU uses the cuSOLVER routines gesvdj and gesvdjBatched on CUDA 10.1.243 and later, and uses the MAGMA routine gesdd on earlier versions of CUDA.

Note

The returned matrix U will be transposed, i.e. with strides U.contiguous().transpose(-2, -1).stride().

Note

Gradients computed using U and Vh may be unstable if input is not full rank or has non-unique singular values.

Note

When full_matrices = True, the gradients on U[..., :, min(m, n):] and V[..., :, min(m, n):] will be ignored in backward as those vectors can be arbitrary bases of the subspaces.

Note

The S tensor can only be used to compute gradients if compute_uv is True.

Note

Since U and V of an SVD is not unique, each vector can be multiplied by an arbitrary phase factor $e^{i \phi}$ while the SVD result is still correct. Different platforms, like Numpy, or inputs on different device types, may produce different U and V tensors.

Parameters
• input (Tensor) – the input tensor of size $(*, m, n)$ where * is zero or more batch dimensions consisting of $m \times n$ matrices.

• full_matrices (bool, optional) – controls whether to compute the full or reduced decomposition, and consequently the shape of returned U and V. Defaults to True.

• compute_uv (bool, optional) – whether to compute U and V or not. Defaults to True.

• out (tuple, optional) – a tuple of three tensors to use for the outputs. If compute_uv=False, the 1st and 3rd arguments must be tensors, but they are ignored. E.g. you can pass (torch.Tensor(), out_S, torch.Tensor())

Example:

>>> import torch
>>> a = torch.randn(5, 3)
>>> a
tensor([[-0.3357, -0.2987, -1.1096],
[ 1.4894,  1.0016, -0.4572],
[-1.9401,  0.7437,  2.0968],
[ 0.1515,  1.3812,  1.5491],
[-1.8489, -0.5907, -2.5673]])
>>>
>>> # reconstruction in the full_matrices=False case
>>> u, s, vh = torch.linalg.svd(a, full_matrices=False)
>>> u.shape, s.shape, vh.shape
(torch.Size([5, 3]), torch.Size([3]), torch.Size([3, 3]))
>>> torch.dist(a, u @ torch.diag(s) @ vh)
tensor(1.0486e-06)
>>>
>>> # reconstruction in the full_matrices=True case
>>> u, s, vh = torch.linalg.svd(a)
>>> u.shape, s.shape, vh.shape
(torch.Size([5, 5]), torch.Size([3]), torch.Size([3, 3]))
>>> torch.dist(a, u[:, :3] @ torch.diag(s) @ vh)
>>> torch.dist(a, u[:, :3] @ torch.diag(s) @ vh)
tensor(1.0486e-06)
>>>
>>> # extra dimensions
>>> a_big = torch.randn(7, 5, 3)
>>> u, s, vh = torch.linalg.svd(a_big, full_matrices=False)
>>> torch.dist(a_big, u @ torch.diag_embed(s) @ vh)
tensor(3.0957e-06)

torch.linalg.solve(input, other, *, out=None) → Tensor

Computes the solution x to the matrix equation matmul(input, x) = other with a square matrix, or batches of such matrices, input and one or more right-hand side vectors other. If input is batched and other is not, then other is broadcast to have the same batch dimensions as input. The resulting tensor has the same shape as the (possibly broadcast) other.

Supports input of float, double, cfloat and cdouble dtypes.

Note

If input is a non-square or non-invertible matrix, or a batch containing non-square matrices or one or more non-invertible matrices, then a RuntimeError will be thrown.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Parameters
• input (Tensor) – the square $n \times n$ matrix or the batch of such matrices of size $(*, n, n)$ where * is one or more batch dimensions.

• other (Tensor) – right-hand side tensor of shape $(*, n)$ or $(*, n, k)$ , where $k$ is the number of right-hand side vectors.

Keyword Arguments

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

Examples:

>>> A = torch.eye(3)
>>> b = torch.randn(3)
>>> x = torch.linalg.solve(A, b)
>>> torch.allclose(A @ x, b)
True


Batched input:

>>> A = torch.randn(2, 3, 3)
>>> b = torch.randn(3, 1)
>>> x = torch.linalg.solve(A, b)
>>> torch.allclose(A @ x, b)
True
>>> b = torch.rand(3) # b is broadcast internally to (*A.shape[:-2], 3)
>>> x = torch.linalg.solve(A, b)
>>> x.shape
torch.Size([2, 3])
>>> Ax = A @ x.unsqueeze(-1)
>>> torch.allclose(Ax, b.unsqueeze(-1).expand_as(Ax))
True

torch.linalg.tensorinv(input, ind=2, *, out=None) → Tensor

Computes a tensor input_inv such that tensordot(input_inv, input, ind) == I_n (inverse tensor equation), where I_n is the n-dimensional identity tensor and n is equal to input.ndim. The resulting tensor input_inv has shape equal to input.shape[ind:] + input.shape[:ind].

Supports input of float, double, cfloat and cdouble data types.

Note

If input is not invertible or does not satisfy the requirement prod(input.shape[ind:]) == prod(input.shape[:ind]), then a RuntimeError will be thrown.

Note

When input is a 2-dimensional tensor and ind=1, this function computes the (multiplicative) inverse of input, equivalent to calling torch.inverse().

Parameters
• input (Tensor) – A tensor to invert. Its shape must satisfy prod(input.shape[:ind]) == prod(input.shape[ind:]).

• ind (int) – A positive integer that describes the inverse tensor equation. See torch.tensordot() for details. Default: 2.

Keyword Arguments

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

Examples:

>>> a = torch.eye(4 * 6).reshape((4, 6, 8, 3))
>>> ainv = torch.linalg.tensorinv(a, ind=2)
>>> ainv.shape
torch.Size([8, 3, 4, 6])
>>> b = torch.randn(4, 6)
>>> torch.allclose(torch.tensordot(ainv, b), torch.linalg.tensorsolve(a, b))
True

>>> a = torch.randn(4, 4)
>>> a_tensorinv = torch.linalg.tensorinv(a, ind=1)
>>> a_inv = torch.inverse(a)
>>> torch.allclose(a_tensorinv, a_inv)
True

torch.linalg.tensorsolve(input, other, dims=None, *, out=None) → Tensor

Computes a tensor x such that tensordot(input, x, dims=x.ndim) = other. The resulting tensor x has the same shape as input[other.ndim:].

Supports real-valued and complex-valued inputs.

Note

If input does not satisfy the requirement prod(input.shape[other.ndim:]) == prod(input.shape[:other.ndim]) after (optionally) moving the dimensions using dims, then a RuntimeError will be thrown.

Parameters
• input (Tensor) – “left-hand-side” tensor, it must satisfy the requirement prod(input.shape[other.ndim:]) == prod(input.shape[:other.ndim]).

• other (Tensor) – “right-hand-side” tensor of shape input.shape[other.ndim].

• dims (Tuple[int]) – dimensions of input to be moved before the computation. Equivalent to calling input = movedim(input, dims, range(len(dims) - input.ndim, 0)). If None (default), no dimensions are moved.

Keyword Arguments

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

Examples:

>>> a = torch.eye(2 * 3 * 4).reshape((2 * 3, 4, 2, 3, 4))
>>> b = torch.randn(2 * 3, 4)
>>> x = torch.linalg.tensorsolve(a, b)
>>> x.shape
torch.Size([2, 3, 4])
>>> torch.allclose(torch.tensordot(a, x, dims=x.ndim), b)
True

>>> a = torch.randn(6, 4, 4, 3, 2)
>>> b = torch.randn(4, 3, 2)
>>> x = torch.linalg.tensorsolve(a, b, dims=(0, 2))
>>> x.shape
torch.Size([6, 4])
>>> a = a.permute(1, 3, 4, 0, 2)
>>> a.shape[b.ndim:]
torch.Size([6, 4])
>>> torch.allclose(torch.tensordot(a, x, dims=x.ndim), b, atol=1e-6)
True

torch.linalg.inv(input, *, out=None) → Tensor

Computes the multiplicative inverse matrix of a square matrix input, or of each square matrix in a batched input. The result satisfies the relation:

matmul(inv(input),input) = matmul(input,inv(input)) = eye(input.shape[0]).expand_as(input).

Supports input of float, double, cfloat and cdouble data types.

Note

When given inputs on a CUDA device, this function synchronizes that device with the CPU.

Note

The inverse matrix is computed using LAPACK’s getrf and getri routines for CPU inputs. For CUDA inputs, cuSOLVER’s getrf and getrs routines as well as cuBLAS’ getrf and getri routines are used if CUDA version >= 10.1.243, otherwise MAGMA’s getrf and getri routines are used instead.

Note

If input is a non-invertible matrix or non-square matrix, or batch with at least one such matrix, then a RuntimeError will be thrown.

Parameters

input (Tensor) – the square (n, n) matrix or the batch of such matrices of size (*, n, n) where * is one or more batch dimensions.

Keyword Arguments

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

Examples:

>>> x = torch.rand(4, 4)
>>> y = torch.linalg.inv(x)
>>> z = torch.mm(x, y)
>>> z
tensor([[ 1.0000, -0.0000, -0.0000,  0.0000],
[ 0.0000,  1.0000,  0.0000,  0.0000],
[ 0.0000,  0.0000,  1.0000,  0.0000],
[ 0.0000, -0.0000, -0.0000,  1.0000]])
>>> torch.max(torch.abs(z - torch.eye(4))) # Max non-zero
tensor(1.1921e-07)

>>> # Batched inverse example
>>> x = torch.randn(2, 3, 4, 4)
>>> y = torch.linalg.inv(x)
>>> z = torch.matmul(x, y)
>>> torch.max(torch.abs(z - torch.eye(4).expand_as(x))) # Max non-zero
tensor(1.9073e-06)

>>> x = torch.rand(4, 4, dtype=torch.cdouble)
>>> y = torch.linalg.inv(x)
>>> z = torch.mm(x, y)
>>> z
tensor([[ 1.0000e+00+0.0000e+00j, -1.3878e-16+3.4694e-16j,
5.5511e-17-1.1102e-16j,  0.0000e+00-1.6653e-16j],
[ 5.5511e-16-1.6653e-16j,  1.0000e+00+6.9389e-17j,
2.2204e-16-1.1102e-16j, -2.2204e-16+1.1102e-16j],
[ 3.8858e-16-1.2490e-16j,  2.7756e-17+3.4694e-17j,
1.0000e+00+0.0000e+00j, -4.4409e-16+5.5511e-17j],
[ 4.4409e-16+5.5511e-16j, -3.8858e-16+1.8041e-16j,
2.2204e-16+0.0000e+00j,  1.0000e+00-3.4694e-16j]],
dtype=torch.complex128)
>>> torch.max(torch.abs(z - torch.eye(4, dtype=torch.cdouble))) # Max non-zero
tensor(7.5107e-16, dtype=torch.float64)

torch.linalg.qr(input, mode='reduced', *, out=None) -> (Tensor, Tensor)

Computes the QR decomposition of a matrix or a batch of matrices input, and returns a namedtuple (Q, R) of tensors such that $\text{input} = Q R$ with $Q$ being an orthogonal matrix or batch of orthogonal matrices and $R$ being an upper triangular matrix or batch of upper triangular matrices.

Depending on the value of mode this function returns the reduced or complete QR factorization. See below for a list of valid modes.

Note

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

Backpropagation is not supported for mode='r'. Use mode='reduced' instead.

Backpropagation is also not supported if the first $\min(input.size(-1), input.size(-2))$ columns of any matrix in input are not linearly independent. While no error will be thrown when this occurs the values of the “gradient” produced may be anything. This behavior may change in the future.

Note

This function uses LAPACK for CPU inputs and MAGMA for CUDA inputs, and may produce different (valid) decompositions on different device types or different platforms.

Parameters
• input (Tensor) – the input tensor of size $(*, m, n)$ where * is zero or more batch dimensions consisting of matrices of dimension $m \times n$ .

• mode (str, optional) –

if k = min(m, n) then:

• 'reduced' : returns (Q, R) with dimensions (m, k), (k, n) (default)

• 'complete': returns (Q, R) with dimensions (m, m), (m, n)

• 'r': computes only R; returns (Q, R) where Q is empty and R has dimensions (k, n)

Keyword Arguments

out (tuple, optional) – tuple of Q and R tensors. The dimensions of Q and R are detailed in the description of mode above.

Example:

>>> 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]])
>>> torch.mm(q, r).round()
tensor([[  12.,  -51.,    4.],
[   6.,  167.,  -68.],
[  -4.,   24.,  -41.]])
>>> torch.mm(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.allclose(torch.matmul(q, r), a)
True
>>> torch.allclose(torch.matmul(q.transpose(-2, -1), q), torch.eye(5))
True