Michael J. Tsatsomeros

Principal pivot transforms: properties and applications

The principal pivot transform (PPT) is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right-hand side of the system. The notion of the PPT is encountered in mathematical programming, statistics and numerical analysis among other areas. The purpose of this paper is to draw attention to the main properties and uses of PPTs, make some new observations and motivate further applications of PPTs in matrix theory. Special consideration is given to PPTs of matrices whose principal minors are positive.

Doubly diagonally dominant matrices

We consider the class of doubly diagonally dominant matrices (A = [ ajj] E C”, ‘, la,,1 l”jjl > Ck+ i laiklCk+ jlajkl. i #j) and its subclasses. We give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an H-matrix. As in the case of diagonal dominance, we show that the Schur complements of doubly diagonally dominant matrices inherit this property. Moreover, we describe when a Schur complement of a strictly doubly diagonally dominant matrix is strictly diagonally dominant. 0 Elsevier Science Inc., 1997 1. PRELIMINARIES

AN ITERATIVE CRITERION FOR H-MATRICES

We provide an algorithmic characterization of H-matrices. When A is an H-matrix, this algorithm determines a positive diagonal matrix D such that AD is strictly row diagonally dominant. In effect, D is produced iteratively by quantifying and

Inverse M-Matrix Inequalities and Generalized Ultrametric Matrices

Abstract We use weighted directed graphs to introduce a class of nonnegative matrices which, under a simple condition, are inverse M-matrices. We call our class the generalized ultrametic matrices, since it contains the class of (symmetric) ultrametric matrices and some unsymmetric matrices. We show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant M-matrix if and only if it contains no zero row and no two of its rows are identical. This theorem generalizes the known result that a (symmetric) strictly ultrametric matrix is the inverse of a strictly diagonally dominant M-matrix. We also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant M-matrix. Some of these inequalities and conditions for equality generalize results of Stieltjes on inverses of symmetric diagonally dominant M-matrices.

CONVEX SETS OF NONSINGULAR AND P-MATRICES

We show that the set r(A,B) (resp. c(A,B)) of square matrices whose rows (resp. columns) are independent convex combinations of

Reachability and Holdability of Nonnegative States

Linear differential systems

Perron–Frobenius type results on the numerical range

\dot{x}(t)=Ax(t)

Ray patterns of matrices and nonsingularity

(

A Recursive Test for P-Matrices

A\in\mathbb{R}^{n\times n}

On the Spectra of Striped Sign Patterns

,