Additional Topics in Linear Algebra

This chapter is devoted to several additional topics in linear algebra and, more specifically, the theory of matrices. In the first section we introduce the notion of a matrix norm and show how such norms can be induced from norms on the spaces Rn and Cn. The second section deals with the Moore-Penrose inverse of a matrix (also called the pseudoinverse). Section three takes up the theory of (real) non-negative matrices, that is, matrices all of whose entries are nonnegative, which has multiple applications. Section four, where we prove the Gers˘gorin disc theorem, deals with the location of eigenvalues of a complex matrix. Finally, in section five we give meaning to the notion of exponentiating a real or complex matrix.