Some results on a partition of Z-matrices Original Research Article

Fiedler and Markham define an n × n matrix A to be an Lk-matrix if A has the form A = tI − B, where B is nonnegative and varrhok(B) less-than-or-equals, slant t < varrhok+1(B), K = 1,2,…, n. Here, varrhok(B) denotes the maximum spectral radius of all principal submatrices of B of order k for k = 1,2,…, n, and for completeness, varrhon+1(B) is defined to be +∞. Further, A is defined to be an L0-matrix if A = tI − B, where B is nonnegative and t < varrho1(B). The classes L0 L1,…, Ln 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 Lk-matrices. Inverse Lk-matrices and Schur complements of Lk-matrices are also studied. An eigenvalue inequality involving Schur complements of Lk-matrices is obtained.