Minerva Catral

BLOCK REPRESENTATIONS OF THE DRAZIN INVERSE OF A BIPARTITE MATRIX

Block representations of the Drazin inverse of a bipartite matrix A = 0 B C 0 in terms of the Drazin inverse of the smaller order block product BC or CBare presented. Relationships between the index of A and the index of BC are determined, and examples are given to illustrate all such possible relationships.

GROUP INVERSES OF MATRICES WITH PATH GRAPHS

A simple formula for the group inverse of a 2 × 2 block matrix with a bipartite digraph is given in terms of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of such matrices are given. An extension of the graph-theoretic description of the group inverse to singular matrices with tree graphs is conjectured.

Sign patterns that allow eventual positivity

Several necessary or sufficient conditions for a sign patternto allow eventual posi- tivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n � 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.

The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains

A quantity known as the Kemeny constant, which is used to measure the expected number of links that a surfer on the World Wide Web, located on a random web page, needs to follow before reaching his/her desired location, coincides with the more well known notion of the expected time to mixing, i.e., to reaching stationarity of an ergodic Markov chain. In this paper we present a new formula for the Kemeny constant and we develop several perturbation results for the constant, including conditions under which it is a convex function. Finally, for chains whose transition matrix has a certain directed graph structure we show that the Kemeny constant is dependent only on the common length of the cycles and the total number of vertices and not on the specific transition probabilities of the chain.

Sign patterns that require or allow power-positivity

A matrix A is power-positive if some positive integer power of A is entrywise positive. A sign pattern A is shown to require power-positivity if and only if either A or A is nonnegative and has a primitive digraph, or equivalently, either A or A requires eventual positivity. A sign pattern A is shown to be potentially power-positive if and only if A or A is potentially eventually positive.

On functions that preserve M-matrices and inverse M-matrices

In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as the Stieltjes matrices. Here we characterize functions which either preserve the inverse M-matrices or map the inverse M-matrices to the M-matrices. In one of our results we employ the theory of Pick functions to show that if A and B are inverse M-matrices such that B −1 ≤ A −1, then (B+tI)−1 ≤ (A+tI)−1, for all t ≥ 0.

MINIMUM RANK, MAXIMUM NULLITY, AND ZERO FORCING NUMBER OF SIMPLE DIGRAPHS

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simple digraphs are established. The effect of deletion of a vertex on minimum rank or zero forcing number is analyzed, and simple digraphs having very low or very high zero forcing number are characterized.

Zero Forcing Propagation Time on Oriented Graphs

Abstract Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing set. In this paper we consider the minimum number of iterations needed for this color change rule to color all of the vertices blue, also known as the propagation time, for oriented graphs. We produce oriented graphs with both high and low propagation times, consider the possible propagation times for the orientations of a fixed graph, and look at balancing the size of a zero forcing set and the propagation time.

Sign patterns that allow strong eventual nonnegativity

A new class of sign patterns contained in the class of sign patterns that allow eventual nonnegativity is introduced and studied. A sign pattern is potentially strongly eventually nonnegative (PSEN) if there is a matrix with this sign pattern that is eventually nonnegative and has some power that is both nonnegative and irreducible. Using Perron-Frobenius theory and a matrix perturbation result, it is proved that a PSEN sign pattern is either potentially eventually positive or r-cyclic. The minimum number of positive entries in an n× n PSEN sign pattern is shown to be n, and PSEN sign patterns of orders 2 and 3 are characterized.

The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

The enhanced principal rank characteristic sequence (epr-sequence) of an