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# torch.linalg¶

Common linear algebra operations.

## 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 :attr: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

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.

Warning

This function always checks whether input is a Hermitian positive-definite matrix using info argument to LAPACK/MAGMA call. For CUDA this causes cross-device memory synchronization.

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.det(input) → Tensor

Alias of torch.det().

torch.linalg.eigh(input, UPLO='L') -> tuple(Tensor, Tensor)

This function computes the eigenvalues and eigenvectors of a complex Hermitian (or real symmetric) matrix, or batch of such matrices, input. For a single matrix input, the tensor of eigenvalues $w$ and the tensor of eigenvectors $V$ decompose the input such that $\text{input} = V \text{diag}(w) V^H$ , where $^H$ is the conjugate transpose operation.

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 data types.

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

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

The eigenvalues/eigenvectors are computed using LAPACK/MAGMA routines _syevd and _heevd. This function always checks whether the call to LAPACK/MAGMA is successful using info argument of _syevd, _heevd and throws a RuntimeError if it isn’t. On CUDA this causes a cross-device memory synchronization.

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: 'L'

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') → Tensor

This function computes the eigenvalues of a complex Hermitian (or real symmetric) matrix, or batch of such matrices, 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 data types.

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

Note

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

Note

The eigenvalues/eigenvectors are computed using LAPACK/MAGMA routines _syevd and _heevd. This function always checks whether the call to LAPACK/MAGMA is successful using info argument of _syevd, _heevd and throws a RuntimeError if it isn’t. On CUDA this causes a cross-device memory synchronization.

Note

This function doesn’t support backpropagation, please use torch.linalg.eigh() instead, that also computes the 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: 'L'

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.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.

• 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.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
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