What is Cholesky Decomposition?
Cholesky Decomposition is the decomposition of Hermitian, positive definite matrix into the multiplication of two matrices, where one is a positive diagonal lower triangular matrix and the other is its conjugate transpose matrix i.e., upper triangular matrix. As Cholesky decomposition can represent matrices as a product of two matrices, it is also called Cholesky Factorization.
The Cholesky decomposition is specially defined for symmetric matrices and Cholesky Decomposition is used widely as it is faster than the LU decomposition. Let A be any Hermitian, positive definite matrix, then the Cholesky decomposition can be given by:
A = LL*
Where,
- L is the lower triangular matrix of A with a positive diagonal
- L* is the conjugate transpose matrix of A
Note: Every Hermitian positive definite matrix has a unique Cholesky decomposition.
Cholesky Decomposition
Cholesky Decomposition is one of the types of many decompositions in linear algebra which is a branch of mathematics that deals with linear equations and vectors. Decomposition is the term related to the factorization of matrices in linear algebra, and Cholesky is one of the ways to factorize or decompose the matrix into two matrices. This article explores the Cholesky Decomposition in detail including its definition, steps to factorize matrices using Cholesky Decomposition, and some of the solved examples. So, let’s start learning about this exciting topic of Cholesky Decomposition.