ABSTRACT
As stated in the Preface, this book assumes an undergraduate knowledge of linear algebra. In this chapter, we study topics that are typically beyond that of an undergraduate linear algebra course, but are useful in later chapters of this book. Much of the material is taken from [6] and [41] which are two standard resources in linear algebra. We begin with a study of vector and matrix norms. Vector and matrix norms are useful in finding bounds on the spectral radius of a square matrix. We study the spectral radius of matrices more extensively in the next section which covers Perron-Frobenius theory. Perron-Frobenius theory is the study of nonnegative matrices. We will study nonnegative matrices in general, but also study interesting subsets of this class of matrices, namely positive matrices and irreducible matrices. We will see that positive matrices and irreducible matrices have many of the same properties. Nonnegative matrices will play an important role throughout this book and will be useful in understanding the theory behind M-matrices which also play an important role in later chapters. Hence we dedicate a section to M-matrices and apply the theory of nonnegative matrices to proofs of theorems involving Mmatrices. Nonnegative matrices are also useful in the study of doubly stochastic matrices. Doubly stochastic matrices, which we study in the section following the section on M-matrices, are nonnegative matrices whose row sums and column sums are each one. Doubly stochastic matrices will play an important role in the study of the algebraic connectivity of graphs. Finally, we close this chapter with a section on generalized inverses of matrices. Since many of the matrices we will utilize in this book are singular, we need to familiarize ourselves with more general inverses, namely the group inverse of matrices.
