Various topics of matrix theory, in particular, those related to nonnegative matrices (matrices with nonnegative entries) are considered in this chapter. We introduce the concepts of reducible and irreducible matrices and matrix graph theory (the concepts of directed and strongly connected graphs), and show the equivalence between irreducibility of a matrix and the connectivity of its directed graph. This enables us, among other things, to strengthen the Gershgorin theorem for estimating the location of eigenvalues of irreducible matrices. In order to determine if a symmetric matrix is positive definite, we need information regarding the signs of its eigenvalues. Also, in order to determine the rate of convergence of certain iterative methods to solve linear systems of algebraic equations, we need to know—as we shall see in later chapters—some information regarding the location of the eigenvalues of the iteration matrix.

The Perron-Frobenius theorem, showing that the spectral radius ρ(A) is an eigenvalue corresponding to a positive eigenvector, if A is nonnegative and irreducible, is presented. It will be seen in some of the following chapters that the concept of numerical radius can give sharper estimates of the norm of the powers of matrices, for instance, than the spectral radius can. Some results relating the numerical radius with the norm of the matrix and with the spectral radius of the symmetric part of nonnegative matrices are presented.