Matrix decomposition: Difference between revisions

Revision as of 04:09, 18 March 2021

Revision as of 22:22, 28 March 2021

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{{main|Cholesky decomposition}}

*Applicable to: [[square matrix|square]], [[symmetric matrix|hermitian]], [[positive-definite matrix|positive definite]] matrix ''A''

*Decomposition: <math>A=U^*U^\mathsf{T}</math>, where <math>U</math> is upper triangular with real positive diagonal entries

*Comment: if the matrix <math>A</math> is Hermitian and positive semi-definite, then it has a decomposition of the form <math>A=U^*U^\mathsf{T}</math> if the diagonal entries of <math>U</math> are allowed to be zero

*Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case.

*Comment: if A is real and symmetric, <math>U</math> has all real elements