Matrix Algebra for MDS

Abstract

In this chapter, we build a basis for a more technical understanding of MDS. Matrices are of particular importance here. They bring together, in one single mathematical object, such notions as a whole configuration of points, all of the distances among the points of this configuration, or a complete matrix of proximities. Mathematicians developed a sophisticated algebra for matrices that allows one to derive, for example, how a configuration that represents a matrix of distances can be computed, or how the distances among all points can be derived from a configuration. Most of these operations can be written in just a few lines, in very compact notation, which helps tremendously to see what is going on. The reader does not have to know everything in this chapter to read on in this book. It suffices to know the main concepts and theorems and then later come back to this chapter when necessary. Proofs in this chapter are meant to better familiarize the reader with the various notions. One may opt to skip the proofs and accept the respective theorems, as is common practice in mathematics (“It can be shown that...”).