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In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful e.g. for efficient numerical solutions and Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations. == Statement == The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form : where L is a lower triangular matrix with real and positive diagonal entries, and L * denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.〔, , 〕 If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL * if the diagonal entries of L are allowed to be zero. When A has real entries, L has real entries as well and the factorization may be written A = LLT The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL *. However, the decomposition need not be unique when A is positive semidefinite. The converse holds trivially: if A can be written as LL * for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cholesky decomposition」の詳細全文を読む スポンサード リンク
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