Cholesky分解は、Choleskyとしても知られています 分解, is a mathematical technique used in 線形代数 to factorize a positive-definite matrix into a product of a lower triangular matrix and its transpose. This factorization is particularly useful in various applications, including solving systems of linear equations, optimization problems, and in 統計的方法.
具体的には、 A is a symmetric, positive-definite matrix, the Cholesky Factorization states that there exists a unique lower triangular matrix L そのような場合：
A = L * L^TT
where LT is the transpose of L. The process of obtaining L involves a series of calculations that eliminate variables step by step, ensuring that the resulting matrix is triangular.
Cholesky Factorization is computationally efficient, requiring approximately half the number of operations needed for other factorization methods such as LU decomposition. Its advantages make it favorable in algorithms requiring matrix inversion or solving linear systems, especially in the context of machine learning and numerical simulations. Additionally, it plays a critical role in Monte Carlo methods and 最適化アルゴリズム, where the efficiency of matrix computations is crucial.
要約すると、Cholesky分解は 数値線形代数において強力なツールです, providing a means to simplify complex calculations involving positive-definite matrices.