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CholeskyDecomposition[m] computes the Cholesky decomposition of a matrix m.
CholeskyDecomposition[m] returns a list of the form lmat, perm, diag, where lmat is a lower-triangular matrix, perm is a permutation vector and diag is a vector corresponding to the leading diagonal of a matrix.
When perm is the identity permutation and diag is a zero vector, then lmat . Transpose[lmat] is exactly the original matrix m.
In general, lmat . Transpose[lmat] is given by Transpose[p] . m . p + DiagonalMatrix[diag] where p = IdentityMatrix[Length[perm]][[perm]].
CholeskyDecomposition works with both numerical and symbolic square matrices.
CholeskyDecomposition regularizes all Hermitian numerical matrices to make them positive definite Hermitian.
See also: LUDecomposition, LUBackSubstitution, LinearSolve.
Note: this is an experimental feature, and in future versions of Mathematica it may not be supported, or may have a different specification.