Completions of Matrices

Abstract

This chapter, which has an introductory character, deals with completion problems for finite matrices with scalar entries. Here we introduce the simplest versions of the problems that will be treated in the next chapters with more powerful tools. This first chapter presents in a nutshell some of the main features of completion problems, and the results obtained here serve as a motivation for the more sophisticated theorems that will follow later. This chapter consists of three sections. In the first section we analyze the possibilities of one step extensions for invertible selfadjoint matrices. In the second section the results are developed further for the positive definite case. Here notions such as band extension and maximum entropy solution appear. In the last section analogous results are derived for contractive completion problems by reduction to the positive definite case.