LDL* Factorization and Positive Definite Matrices

Abstract

In this chapter we consider LU factorizations of Hermitian and positive definite matrices. Recall that a matrix ${\boldsymbol {A}}\in {\mathbb {C}}^{n\times n}$ is Hermitian if A ^∗ = A, i.e., $a_{ji}=\overline {a}_{ij}$ for all i, j. A real Hermitian matrix is symmetric. Since $a_{ii}=\overline {a}_{ii}$ the diagonal elements of a Hermitian matrix must be real.