Introduction

In this chapter, first we provide a brief overview of some of the advances that have been made recently in the theory of infinite matrices and their applications. Then we include a summary of the contents of each chapter.Infinite matrices have applications in many branches of classical mathematics such as infinite quadratic forms, integral equations, and differential equations. As illustrated in Chap. 8 , this topic has applications in other sciences besides mathematics, as well. A review of some of the topics of this monograph was recently undertaken by Shivakumar and Sivakumar [111]. In this monograph, apart from elaborating on some of the topics that are discussed in that review, we include other interesting topics such as quaternion matrices and infinite dimensional extensions of certain positivity classes of matrices.Gaussian elimination, the familiar method for solving systems of finitely many linear equations in finitely many unknowns, has been around for over two hundred years. Unlike the matrix methods, matrices were not used in the early formulations of Gaussian elimination, until the mid-twentieth century.