Daniel Hershkowitz

Recent directions in matrix stability

Abstract Matrix stability has been intensively investigated in the past two centuries. We review work that has been done in this topic, focusing on the great progress that has been achieved in the last decade or two. We start with classical stability criteria of Lyapunov, Routh and Hurwitz, and Lienard and Chipart. We then study recently proven sufficient conditions for stability, with particular emphasis on P -matrices. We investigate conditions for the existence of a stable scaling for a given matrix. We review results on other types of matrix stability, such as D -stability, additive D -stability, and Lyapunov diagonal stability. We discuss the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices. We also discuss the uniqueness of Lyapunov scaling factors, maximal Lyapunov scaling factors, cones of real positive semidefinite matrices and their applications to matrix stability, and inertia preserving matrices.

Ranks of zero patterns and sign patterns

Let F be a field with at least three elements. Zero patterns P such that all matrices over F with pattern P have the same rank are characterized. Similar results are proven for sign patterns. These results are applied to answering two open questions on conditions for formal nonsingularity of a pattern P, as well as to proving a sufficient condition on P such that all matrices over F with pattern P have the same height characteristic.

Matrix Diagonal Stability and Its Implications

Relations between diagonal stability, stability, positiveness of principal minors and semipositivity are described for several classes of matrices. In particular, it is shown that for matrices whose nondirected graph is acyclic, positiveness of principal minors is equivalent to diagonal stability.

The spread of the spectrum of a graph

Upper and lower bounds are obtained for the spread λ1 − λn of the eigenvalues λ1 λ2 ··· λn of the adjacency matrix of a simple graph.

Combinatorial results on completely positive matrices

Abstract A sufficient condition for complete positivity of a matrix, in terms of complete positivity of smaller matrices, is given. Those patterns for which this condition is also necessary are characterized. These results are used to characterize complete positivity of matrices under some acyclicity assumptions on their graph. The exact value of the factorization index is given for acyche matrices.

ON THE MINIMUM RANK OF NOT NECESSARILY SYMMETRIC MATRICES: A PRELIMINARY STUDY ∗

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

Existence of matrices with prescribed eigenvalues and entries

It is proved improving a previous result of Oliveira that apart from two exceptions there always exists an n×n matrix with arbitarily prescribed 2n−3 entries and spectrum. Moreover, it is shown that the number 2n 3 of prescribed entries cannot be increased.

On the generalized nullspace of M-matrices and Z-matrices☆

A proof is given for the preferred basis theorem for the generalized nullspace of a given M-matrix. The theorem is then generalized and extended to the case of a Z-matrix.

Lyapunov diagonal semistability of real H-matrices

We characterize Lyapunov diagonally stable real H-matrices and those real H-matrices which are Lyapunov diagonally semistable but not Lyapunov diagonally stable (called Lyapunov diagonally near-stable). The latter characterization is given in terms of the principal submatrix rank property defined here. We apply our results to the numerical abscissas of real matrices. One of our main tools is a slight strengthening of classical results of Ostrowski which we derive from a fundamental theorem of Wielandt.

On the existence of matrices with prescribed height and level characteristics

We determine all possible relations between the height (Weyr) characteristic and the level characteristic of anM-matrix. Under the assumption that the two characteristics have the same number of elements, we determine the possible relations between the two characteristics for a wider class of matrices, which also contains the class of strictly triangular matrices over an arbitrary field. Given two sequences which satisfy the above condition, we construct a loopless acyclic graphG with the following property: Every matrix whose graph isG has its height characteristic equal to the first sequence and its level characteristic equal to the second. We give several counterexamples to possible extensions of our results, and we raise some open problems.