torch.linalg.cholesky — PyTorch 2.7 documentation (original) (raw)

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

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

Letting K\mathbb{K} be R\mathbb{R} or C\mathbb{C}, the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrixA∈Kn×nA \in \mathbb{K}^{n \times n} is defined as

A=LLHL∈Kn×nA = LL^{\text{H}}\mathrlap{\qquad L \in \mathbb{K}^{n \times n}}

where LL is a lower triangular matrix with real positive diagonal (even in the complex case) andLHL^{\text{H}} is the conjugate transpose when LL is complex, and the transpose when LL 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. For a version of this function that does not synchronize, see torch.linalg.cholesky_ex().

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)