How to Evaluate Cholesky Decomposition?
To decompose a matrix X using Cholesky decomposition we first decompose the matrix A in the form A = LL* where L is assumed to be a positive diagonal lower triangular matrix and L* is the conjugate transpose of L. Then, we have to find the elements of L. To find the value of the diagonal elements we use the formula
Lvv = √(Avv – ∑u<v Lvu (Lvu)*)
And to find the value of non-diagonal elements we use the formula
Ltv = (1 / Lvv)(Atv – ∑u<v Ltu (Lvu)*)
After finding all the elements arrange them in the lower triangular matrix L and then find L* which is the conjugate transpose of L. Finally, evaluate the product of LL* to get the simplified matrix of A.
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.