ABSTRACT
Matrices with special structure — such as diagonal matrices, triangular matrices, and unitary matrices — are simpler to work with than general matrices. Many algorithms have been developed in numerical linear algebra to convert an input matrix into a form with specified special structure by using a sequence of carefully chosen matrix operations. These algorithms can often be described mathematically as providing factorizations of matrices into products of structured matrices. This chapter proves the existence and uniqueness properties of several matrix factorizations and explores the algebraic and geometric ideas leading to these factorizations.
