ABSTRACT

This chapter provides a useful mathematical tool, namely Majorization Theory, and illustrates its applications in a variety of scenarios in signal processing and communication systems. Majorization is a partial ordering and precisely defines the vague notion that the components of a vector are "less spread out" or "more nearly equal" than the components of another vector. Functions that preserve the ordering of majorization are said to be Schur-convex or Schur-concave. Many problems arising in signal processing and communications involve comparing vector-valued strategies or solving optimization problems with vector- or matrix-valued variables. Majorization theory is a key tool that allows us to solve or simplify these problems. The chapter examines the basic concepts and results on majorization that serve mostly the problems in signal processing and communications, but by no means to enclose the vast literature on majorization theory. It devotes to building the framework of majorization theory.