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CholeskyDecomposition

Returns the upper-triangular Cholesky factor U of a Hermitian positive-definite matrix such that ConjugateTranspose[U].U equals the input.

$ wo 'CholeskyDecomposition[{{4, 2}, {2, 2}}]'
{{2, 1}, {0, 1}}

Exact arithmetic keeps radicals symbolic:

$ wo 'CholeskyDecomposition[{{4, 2}, {2, 3}}]'
{{2, 1}, {0, Sqrt[2]}}

Machine-precision input yields a numeric factor:

$ wo 'CholeskyDecomposition[{{2.0, 1.0}, {1.0, 2.0}}]'
{{1.4142135623730951, 0.7071067811865475}, {0., 1.224744871391589}}

Complex Hermitian matrices conjugate the pivot-row factors:

$ wo 'CholeskyDecomposition[{{2, I}, {-I, 2}}]'
{{Sqrt[2], I/Sqrt[2]}, {0, Sqrt[3/2]}}

TargetStructure -> "Structured" wraps the factor as a structured array:

$ wo 'CholeskyDecomposition[{{4, 2}, {2, 3}}, TargetStructure -> "Structured"]'
UpperTriangularMatrix[StructuredArray`StructuredData[{2, 2}, {{2, 1}, {0, Sqrt[2]}}]]