chapter  8
20 Pages

Nonnegative Matrices

In this chapter we consider matrices each of whose entries is a nonnegative number. These matrices have special spectral properties that depend solely on the digraph of the matrix and are independent of the magnitude of the positive entries. Some important classes of nonnegative matrices, such as irreducible (Section 8.1), primitive, and imprimitive matrices (Section 8.2), are defined here, contrary to the standard approach, by properties of associated digraphs (strong connectednes, lengths of cycles etc.). We discuss, mostly without proof, many of the results of the socalled Perron-Frobenius theory of nonnegative matrices (Section 8.3). This theory represents a basic ingredient of the theory of graph spectra where tools from matrix theory are used to study graphs (a direction quite opposite from our main stream here since we want to show how graphs are used to treat matrices). Section 8.4 represents a short introduction to graph spectra.