Z-Pencils

The matrix pencil (A,B) = {tB-A | t \in C} is considered under the assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s introduced by Fiedler and Markham. As no combinatorial structure of B is assumed here, this partition generalizes some of their work where B=I. Based on the union of the directed graphs of A and B, the combinatorial structure of nonnegative eigenvectors associated with the largest eigenvalue of (A,B) in [0,1) is considered.Comment: 8 pages, LaTe

ON REDUCING AND DEFLATING SUBSPACES OF MATRICES

A multilinear approach based on Grassmann representatives and matrix compounds is presented for the identification of reducing pairs of subspaces that are common to two or more matrices. Similar methods are employed to characterize the deflatingpairs of subspaces for a regular matrix pencil A + sB, namely, pairs of subspaces (L, M) such that AL ⊆ M and BL ⊆ M

On the Brualdi-Li Matrix and its Perron Eigenspace

Abstract. The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) ρ among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimating ρ and motivates us to study the Perron eigenvector x of Bn, which is normalized to have 1-norm equal to one. It follows that x minimizes the 2-norm among all Perron vectors of n × n tournament matrices. There are also interesting relations among the entries of x and ρ, allowing us to rank the teams corresponding to a Brualdi-Li tournament according to the Kendall-Wei and Ramanajucharyula ranking schemes