CHAPTER 6 – M-MATRICES

Publisher Summary This chapter discusses the matrices of the form that occur in relation to systems of linear or nonlinear equations or eigenvalue problems in a wide variety of areas including finite difference methods for partial differential equations, input–output production, growth models in economics, linear complementarity problems in operations research, and Markov processes in probability and statistics. The chapter presents a systematic treatment of a certain subclass of matrices in Zn×n called M-matrices. The theory of completely monotonic functions is used to study the inverse-positive property of nonsingular M-matrices. The chapter discusses characterization theorems and M-matrices leading to semiconvergent splittings. These M-matrices arise in solving sparse singular systems of linear equations. The chapter also describes the matrix version of the Neumann lemma for convergent series. For practical purposes in characterizing nonsingular M-matrices, it is evident that one can begin by assuming that AɛZnxn. However, many of the statements of these characterizations are equivalent without this assumption.