torch.linalg.cholesky

torch.linalg.cholesky(A, *, upper=False, out=None) → Tensor

Computes the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix $A \in \mathbb{K}^{n \times n}$ is defined as

$$A = LL^{\text{H}}\mathrlap{\qquad L \in \mathbb{K}^{n \times n}}$$

where $L$ is a lower triangular matrix with real positive diagonal (even in the complex case) and $L^{\text{H}}$ is the conjugate transpose when $L$ is complex, and the transpose when $L$ is real-valued.

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.

Note

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

See also

torch.linalg.cholesky_ex() for a version of this operation that skips the (slow) error checking by default and instead returns the debug information. This makes it a faster way to check if a matrix is positive-definite.

torch.linalg.eigh() for a different decomposition of a Hermitian matrix. The eigenvalue decomposition gives more information about the matrix but it slower to compute than the Cholesky decomposition.

Parameters

A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of symmetric or Hermitian positive-definite matrices.

Keyword Arguments

• upper (bool, optional) – whether to return an upper triangular matrix. The tensor returned with upper=True is the conjugate transpose of the tensor returned with upper=False.

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

Raises

RuntimeError – if the A matrix or any matrix in a batched A is not Hermitian (resp. symmetric) positive-definite. If A is a batch of matrices, the error message will include the batch index of the first matrix that fails to meet this condition.

Examples:

>>> A = torch.randn(2, 2, dtype=torch.complex128)
>>> A = A @ A.T.conj() + torch.eye(2) # creates a Hermitian positive-definite matrix
>>> A
tensor([[2.5266+0.0000j, 1.9586-2.0626j],
        [1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)
>>> L = torch.linalg.cholesky(A)
>>> L
tensor([[1.5895+0.0000j, 0.0000+0.0000j],
        [1.2322+1.2976j, 2.4928+0.0000j]], dtype=torch.complex128)
>>> torch.dist(L @ L.T.conj(), A)
tensor(4.4692e-16, dtype=torch.float64)

>>> A = torch.randn(3, 2, 2, dtype=torch.float64)
>>> A = A @ A.mT + torch.eye(2)  # batch of symmetric positive-definite matrices
>>> L = torch.linalg.cholesky(A)
>>> torch.dist(L @ L.mT, A)
tensor(5.8747e-16, dtype=torch.float64)