Sequences of lower and upper bounds for the spectral radius of a nonnegative matrix

In this article, we expand a classical Frobenius' result upon consecutive k-th powers of nonnegative matrices to establish sequences of new lower and upper bounds for the spectral radius with respect to the positive integer k, each term of which is formulated by the average (k + 1)-row sums of the nonnegative matrix. With the aid of the average (k + 1)-row sums and taking the extreme entries of the matrix, we study new bounds generalizing existing formulae and we produce sequences of new tighter approximations for the spectral radius. The monotonicity and convergence properties of the constructed sequences are explored and certain conditions are stated under which the new bounds are sharper than Frobenius' bounds and other existing formulae. We further characterize the cases of equality in the aforesaid bounds, when the matrix is irreducible. Throughout, we perform illustrative numerical examples to showcase the efficiency of our proposed bounds and make comparisons among them.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

In this article, we present new lower and upper bounds for the spectral radius of nonnegative matrices based on consecutive k-th powers. These bounds are formulated using the average (k + 1)-row sums and extreme entries of the matrix and offer tighter approximations. We analyze the properties of these bounds, including monotonicity and convergence, and provide conditions under which they are sharper than existing formulae. We also explore the cases of equality in the bounds for irreducible matrices and provide illustrative numerical examples.