torch.linalg.ldl_factor(A, *, hermitian=False, out=None)

Computes a compact representation of the LDL factorization of a Hermitian or symmetric (possibly indefinite) matrix.

When A is complex valued it can be Hermitian (hermitian= True) or symmetric (hermitian= False).

The factorization is of the form the form A=LDLTA = L D L^T. If hermitian is True then transpose operation is the conjugate transpose.

LL (or UU) and DD are stored in compact form in LD. They follow the format specified by LAPACK’s sytrf function. These tensors may be used in torch.linalg.ldl_solve() to solve linear systems.

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.


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.ldl_factor_ex().


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

Keyword Arguments:
  • hermitian (bool, optional) – whether to consider the input to be Hermitian or symmetric. For real-valued matrices, this switch has no effect. Default: False.

  • out (tuple, optional) – tuple of two tensors to write the output to. Ignored if None. Default: None.


A named tuple (LD, pivots).


>>> A = torch.randn(3, 3)
>>> A = A @ A.mT # make symmetric
>>> A
tensor([[7.2079, 4.2414, 1.9428],
        [4.2414, 3.4554, 0.3264],
        [1.9428, 0.3264, 1.3823]])
>>> LD, pivots = torch.linalg.ldl_factor(A)
>>> LD
tensor([[ 7.2079,  0.0000,  0.0000],
        [ 0.5884,  0.9595,  0.0000],
        [ 0.2695, -0.8513,  0.1633]])
>>> pivots
tensor([1, 2, 3], dtype=torch.int32)


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