Luz M. DeAlba

Completions of P-matrix patterns

Abstract A list of positions in an n × n real matrix (a pattern) is said to have M - completion if every partial M -matrix that specifies exactly these positions can be completed to an M -matrix. Let Q be a pattern that includes all diagonal entries and let G be its digraph. The following are equivalent. (1) the pattern Q has M -completion; (2) the pattern Q is permutation similar to a block triangular pattern with all the diagonal blocks completely specified; (3) any strongly connected subdigraph of G is complete. A pattern with some diagonal entries unspecified has M -completion if and only if the principal subpattern defined by the specified diagonal positions has M -completion.

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.

UNIVERSALLY OPTIMAL MATRICES AND FIELD INDEPENDENCE OF THE MINIMUM RANK OF A GRAPH

The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry (for ij) isnonzero whenever {i, j} isan edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is0, 1, or −1, and for all fields F , the rank of A isthe minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerousgraphs . Examplesare als o provided verifying lack of field independence for other graphs.

Multiplicative perturbations of stable and convergent operators

Abstract Some familiar classes of stable Hilbert-space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie. Analogous, but less familiar, classes of convergent operators are examined with the same aim. The classes considered are often sets of products M A where M is a given set of diagonal or Hermitian matrices and A is a single matrix. The As for which M A is a set of stable or convergent operators are sometimes characterized.

Possible inertia combinations in the Stein and Lyapunov equations

Abstract In the Stein (or, equivalently, the Lyapunov) equation, we show that the only joint constraints on the inertias of the three matrices are the classical definiteness and semidefiniteness constraints and a simple rank-related constraint.

The (weakly) sign symmetric P-matrix completion problems

In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric P - matrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partialweakl y sign symmetric P0-matrix. Patterns of entries for 4 × 4 matrices are classified as to whether or not a partial (weakly) sign symmetric P - or weakly sign symmetric P0-matrix specifying the pattern must have completion to the same type of matrix. The relationship between the weakly sign symmetric P - and sign symmetric P -matrix completion problems is also examined.

The P_0-matrix completion problem

In this paper the P0-matrix completion problem is considered. It is established that every asymmetric partial P0-matrix has P0-completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P0-completion or not having P0-completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5h asP0-completion.

The nonnegative P_0-matrix completion problem

In this paperthe nonnegative P0-matrix completion problem is considered. It is shown that a pattern for 4 × 4 matrices that includes all diagonal positions has nonnegative P0- completion if and only if its digraph is complete when it has a 4-cycle. It is also shown that any positionally symmetric pattern that includes all diagonal positions and whose graph is an n-cycle has nonnegative P0-completion if and only if n � .

The Q-matrix completion problem

Ar ealn × n matrix is a Q-matrix if for every k =1 , 2,...,nthe sum of all k × k principal minors is positive. A digraph D is said to have Q-completion if every partial Q-matrix specifying D can be completed to a Q-matrix. For the Q-completion problem, sufficient conditions for a digraph to have Q-completion are given, necessary conditions for a digraph to have Q-completion are provided, and those digraphs of order at most four that have Q-completion are characterized.

Inertia of the stein transformation with respect to some nonderogatory matrices

Let A be an n-by-n nonderogatory matrix all of whose eigenvalues lie on the unit circle, and let π and ν be nonnegative integers with π + ν = n. Let π′ and ν′ be positive integers and δ′ a nonnegative integer with π′ + ν′ + δ′ = n. In this paper we explore the existence of a Hermitian nonsingular matrix K with inertia (π, ν, 0), such that the Stein transformation of K corresponding to A, SA(K) = K − AKA∗, is a Hermitian matrix with inertia (π′, ν′, δ′). The study is done by reducing A to Jordan canonical form. If C is an n-by-n nonderogatory matrix all of whose eigenvalues lie on the imaginary axis, then the results obtained for SA(K) are valid for the Lyapunov transformation, LC(K) = CK + KC∗, of K corresponding to C.