Matrix Decompositions
Gérard Favier · 2021
This chapter provides an overview of the most important matrix decompositions, with a more detailed presentation of the eigenvalue decomposition and singular value decomposition (SVD), as well as some of their applications. It presents several results relating to the SVD as the links between SVD and the fundamental spaces of a matrix and certain matrix norms. The chapter establishes the connection between SVD and principal component analysis (PCA), with an application to data compression by reducing the dimensionality of a data matrix. PCA is a widely used method for data analysis and compression. This method is one of the techniques of factor analysis. The chapter describes the CUR decomposition of a matrix, which is based on selecting certain columns and rows. It shows how the SVD can be used to solve the blind source separation problem in an instantaneous linear mixture, first for the noiseless case, and then for the noisy case.