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Showing content from https://docs.pytorch.org/docs/stable/generated/torch.cholesky.html below:

torch.cholesky — PyTorch 2.8 documentation

Computes the Cholesky decomposition of a symmetric positive-definite matrix A A A or for batches of symmetric positive-definite matrices.

If upper is True, the returned matrix U is upper-triangular, and the decomposition has the form:

A = UT U A = U^TU A=UTU

If upper is False, the returned matrix L is lower-triangular, and the decomposition has the form:

A = L LT A = LL^T A=LLT

If upper is True, and A A A is a batch of symmetric positive-definite matrices, then the returned tensor will be composed of upper-triangular Cholesky factors of each of the individual matrices. Similarly, when upper is False, the returned tensor will be composed of lower-triangular Cholesky factors of each of the individual matrices.

Warning

torch.cholesky() is deprecated in favor of torch.linalg.cholesky() and will be removed in a future PyTorch release.

L = torch.cholesky(A) should be replaced with

L = torch.linalg.cholesky(A)

U = torch.cholesky(A, upper=True) should be replaced with

U = torch.linalg.cholesky(A).mH

This transform will produce equivalent results for all valid (symmetric positive definite) inputs.

Parameters

• input (Tensor) – the input tensor A A A of size ( ∗ , n , n ) (*, n, n) (∗,n,n) where * is zero or more batch dimensions consisting of symmetric positive-definite matrices.

• upper (bool, optional) – flag that indicates whether to return a upper or lower triangular matrix. Default: False

Keyword Arguments

out (Tensor, optional) – the output matrix

Example:

>>> a = torch.randn(3, 3)
>>> a = a @ a.mT + 1e-3 # make symmetric positive-definite
>>> l = torch.cholesky(a)
>>> a
tensor([[ 2.4112, -0.7486,  1.4551],
        [-0.7486,  1.3544,  0.1294],
        [ 1.4551,  0.1294,  1.6724]])
>>> l
tensor([[ 1.5528,  0.0000,  0.0000],
        [-0.4821,  1.0592,  0.0000],
        [ 0.9371,  0.5487,  0.7023]])
>>> l @ l.mT
tensor([[ 2.4112, -0.7486,  1.4551],
        [-0.7486,  1.3544,  0.1294],
        [ 1.4551,  0.1294,  1.6724]])
>>> a = torch.randn(3, 2, 2) # Example for batched input
>>> a = a @ a.mT + 1e-03 # make symmetric positive-definite
>>> l = torch.cholesky(a)
>>> z = l @ l.mT
>>> torch.dist(z, a)
tensor(2.3842e-07)

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