Eigenvalue location for nonnegative and Z-matrices

Abstract

Let L k 0 denote the class of n × n Z-matrices A = tl − B with B ⩾ 0 and ϱ k ( B) ⩽ t < ϱ k + 1 ( B), where ϱ k ( B) denotes the maximum spectral radius of k × k principal submatrices of B. Bounds are determined on the number of eigenvalues with positive real parts for A ϵ L k 0, where k satisfies, ⌊ n 2 ⌋ ⩽ k ⩽ n − 1 . For these classes, when k = n − 1 and n − 2, wedges are identified that contain only the unqiue negative eigenvalue of A. These results lead to new eigenvalue location regions for nonnegative matrices.