Some results on a partition of Z-matrices

Abstract

Fiedler and Markham define an n × n matrix A to be an L k -matrix if A has the form A = tI − B, where B is nonnegative and ϱ k ( B) ⩽ t < ϱ k+1 ( B), k = 1,2,…, n. Here, ϱ k ( B) denotes the maximum spectral radius of all principal submatrices of B of order k for k = 1,2,…, n, and for completeness, ϱ n+1 ( B) is defined to be +∞. Further, A is defined to be an L 0-matrix if A = tI − B, where B is nonnegative and t < ϱ 1( B). The classes L 0 L 1,…, L n form a partition of the class of Z-matrices. In this paper, we characterize nonsingular matrices in these classes in terms of the principal minors of their inverses and extend this characterization to general L k -matrices. Inverse L k -matrices and Schur complements of L k -matrices are also studied. An eigenvalue inequality involving Schur complements of L k -matrices is obtained.