ALAFF The Cholesky Factorization Theorem

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Subsection 5.4.2 The Cholesky Factorization Theorem

We will prove the following theorem in Subsection 5.4.4.

Theorem 5.4.2.1. Cholesky Factorization Theorem.

Given an HPD matrix \(A \) there exists a lower triangular matrix \(L \) such that \(A = L L^H \text{.}\) If the diagonal elements of \(L \) are restricted to be positive, \(L \) is unique.

Obviously, there similarly exists an upper triangular matrix \(U \) such that \(A = U^H U \) since we can choose \(U^H = L \text{.}\)

The lower triangular matrix \(L \) is known as the Cholesky factor and \(L L^H \) is known as the Cholesky factorization of \(A \text{.}\) It is unique if the diagonal elements of \(L \) are restricted to be positive. Typically, only the lower (or upper) triangular part of \(A \) is stored, and it is that part that is then overwritten with the result. In our discussions, we will assume that the lower triangular part of \(A \) is stored and overwritten.