Chapter 1:general introduction

One of the most active and continuing subjects in matrix theory during the past century is the study of those linear operators on matrices that leave certain properties or relations of subsets invariant. Such questions are usually called "Linear Preserver Problems." The earliest papers in our reference list are [Frobenius, 1897] and [Kantor, 1897]. Since much effort has been devoted to this type of problem, there have been several excellent survey papers, e.g., [Marcus, 1962], [Marcus, 1971], and [Grone, 1976]. Since the time of Grone's thesis, there have been many developments, encouraged by the introduction of new techniques, many from outside matrix theory itself. We hope this monograph will provide a useful update and convenient reference for the reader. In this introduction, we describe some typical questions and results on linear preserver problems, give some motivation for the study of the subject and finally provide some guidance fur the use of the monograph.