Structured eigenvectors, interlacing, and matrix completions

This dissertation presents results from three areas of applicable matrix analysis: structured eigenvectors, interlacing, and matrix completion problems. Although these are distinct topics, the structured eigenvector results provide connections.;It is a straightforward matrix calculation that if {dollar}\lambda{dollar} is an eigenvalue of A, x an associated structured eigenvector and {dollar}\alpha{dollar} the set of positions in which x has nonzero entries, then {dollar}\lambda{dollar} is also an eigenvalue of the submatrix of A that lies in the rows and columns indexed by {dollar}\alpha{dollar}. We present a converse to this statement and apply the results to interlacing and to matrix completion problems. Several corollaries are obtained that lead to results concerning the case of equality in the interlacing inequalities for Hermitian matrices, and to the problem of the relationship among eigenvalue multiplicities in various submatrices.;Classical interlacing for an Hermitian matrix A may be viewed as describing how many eigenvalues of A must be captured by intervals determined by eigenvalues of a principal submatrix of A. We generalize the classical interlacing theorems by using singular values of off-diagonal blocks of A to construct extended intervals that capture a larger number of eigenvalues. The union of pairs of intervals is also discussed, and applications are mentioned.;The matrix completion results that we present include: the positive semidefinite cycle completion problem for matrices with data from the complex numbers, distance matrix cycle completability conditions, the P-matrix completion problem, and the totally non-negative completion problem. We show that the positive semidefinite cycle completion problem for matrices with complex data is a special case of a larger real positive semidefinite completion problem. In addition, we characterize those graphs for which the cycle conditions on all minimal cycles imply that a partial distance matrix has a distance matrix completion. We also prove that every combinatorially symmetric partial P-matrix has a P-matrix completion and we characterize the class of graphs for which every partial totally nonnegative matrix has a totally nonnegative completion. The structured eigenvector results are used to give a new proof of the the maximum minimum eigenvalue problem for partial Hermitian matrices with a chordal graph

The combinatorially symmetric \u3cem\u3eP\u3c/em\u3e-matrix completion problem

An n-by-n real matrix is called a P-matrix if all its principal minors are positive. The P-matrix completion problem asks which partial P-matrices have a completion to a P-matrix. Here, we prove that every partial P-matrix with combinatorially symmetric specified entries has a P-matrix completion. The general case, in which the combinatorial symmetry assumption is relaxed, is also discussed

Conditions for a totally positive completion in the case of a symmetrically placed cycle

In earlier work, the labelled graphs G for which every combinatorially symmetric totally nonnegative matrix, the graph of whose specified entries is G, has a totally nonnegative completion were identified. For other graphs, additional conditions on the specified data must hold. Here, necessary and sufficient conditions on the specified data, when G is a cycle, are given for both the totally nonnegative and the totally positive completion problems

Principal Submatrices, Geometric Multiplicities, and Structured Eigenvectors

It is a straightforward matrix calculation that if λ is an eigenvalue of A, x an associated eigenvector and α the set of positions in which x has nonzero entries, then λ is also an eigenvalue of the submatrix of A that lies in the rows and columns indexed by α. A converse is presented that is the most general possible in terms of the data we use. Several corollaries are obtained by applying the main result to normal and Hermitian matrices. These corollaries lead to results concerning the case of equality in the interlacing inequalities for Hermitian matrices, and to the problem of the relationship among eigenvalue multiplicities in various principal submatrices

Maximum nullity and zero forcing of circulant graphs

The zero forcing number of a graph has been applied to communication complexity, electrical powergrid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of agraph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterizethe zero forcing number of various circulant graphs, including families of bipartite circulants, as well as allcubic circulants. We extend the de nition of the Möbius ladder to a type of torus product to obtain boundson the minimum rank and the maximum nullity on these products. We obtain equality for torus products byemploying orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined thesenumbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds forall circulant graphs

CONDITIONS FOR A TOTALLY POSITIVE COMPLETION IN THE CASE OF A SYMMETRICALLY PLACED CYCLE ∗

Abstract. In earlier work, the labelled graphs G for which every combinatorially symmetric totally nonnegative matrix, the graph of whose specified entries is G, has a totally nonnegative completion were identified. For other graphs, additional conditions on the specified data must hold. Here, necessary and sufficient conditions on the specified data, when G is a cycle, are given for both the totally nonnegative and the totally positive completion problems. Key words. Totally nonnegative matrices, Totally positive matrices, Partial matrix, Matrix completion problem, Cycles. AMS subject classifications. 15A48, 15A37. 1. Introduction. A matrix is totally positive (nonnegative) if all of its minors, principal or otherwise, are positive (respectively, nonnegative). Totally positive (TP) matrices and totally nonnegative (TN) matrices arise in a variety of applications including splines, statistics, and dynamical systems. A partial matrix is a rectangular array, in which some entries are specified while the remainder are free to be chose

The totally nonnegative completion problem

An n-by-n real matrix is said to be totally positive (nonnegative) if every minor (principal and non-principal) is positive (nonnegative). The totally nonnegative completion problem asks which partially totally nonnegative matrices have a completion to a totally nonnegative matrix. Here we settle the first natural question: for which (labeled) graphs G does every partial totally nonnegative matrix, the graph of whose specified entries is G, have a totally nonnegative completion? Just as in the positive definite case this must play a key role in any further development of the theory

ELA THE COMBINATORIALLY SYMMETRIC P-MATRIX COMPLETION PROBLEM

An n-by-n real matrix is called a P-matrix if all its principal minors are positive. The P-matrix completion problem asks which partial P-matrices have a completion to a P-matrix. Here, we prove that every partial P-matrix with combinatorially symmetric speci ed entries has a P-matrix completion. The general case, in which the combinatorial symmetry assumption is relaxed, is also discussed. Key words. P-matrix, completion problem, combinatorial symmetry AMS(MOS) subject classi cation. 15A48 An n-by-n real matrix is called a P-matrix (P0-matrix) if all its principal minors are positive (nonnegative), see, e.g., [HJ2] for a brief discussion of these classical notions. This class generalizes many other important classes of matrices (such as positive de nite, M-matrices, and totally positive), has useful structure (such asinverse closure, inheritance by principal submatrices, and wedge type eigenvalue restrictions), and arises in applications (such asthe linear complementarity problem, and issues of local invertibility of functions). A partial matrix is a rectangular array in which some entries are speci ed