New Bounds on Spectral Radius and Minimum Eigenvalue of Structured Matrices

Abstract

The eigenvalue problem of structured matrices has always been a significant research topic in matrix analysis. In this paper, we present a new upper bound on the spectral radius of nonnegative matrices involving the Hadamard product, which generalizes and improves some existing ones. For two $n\times n$ nonsingular $M$-matrices $A$ and $B$, some new lower bounds for the minimum eigenvalue related to the Fan product of $A$ and $B$ are obtained, and meanwhile, the detailed analysis and theoretical comparison between the newly proposed lower bounds and some existing results are also investigated. These estimations that only depend on the entries of the given matrices are not difficult to implement. To further illustrate the effectiveness of the main results, some numerical examples are given.