Nonnegative Matrices

In investigations of the rapidly of convergence of various iterative methods, the spectral radius ρ(A), we have thus far only the upper bounds for ρ(A) of Sect. 1.4, provided by extensions of Gerschgorin's Theorem 1.11. In this section, we shall look closely into the Perron-Frobenius theory of square matrices having nonnegative real numbers as entries. Not only will this theory provide us with both nontrivial upper and lower bounds for the spectral radius for this class of matrices, but the structure of this theory will be decidedly germane to our subsequent development of iterative methods.