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

Inheritance of Striga hermonthica adaptive traits in an earlymaturing white maize inbred line containing resistance genes from Zea diploperennis

Striga hermonthica can cause as high as 100% yield loss in maize depending on soil fertility level, type of genotype, severity of infestation and climatic conditions. Understanding the mode of inheritance of Striga resistance in maize is crucial for introgression of resistance genes into tropical germplasm and deployment of resistant varieties. This study examined the mode of inheritance of resistance to Striga in early‐maturing inbred line, TZdEI 352 containing resistance genes from Zea diploperennis. Six generations, P1, P2, F1, F2, BC1P1 and BC1P2 derived from a cross between resistant line, TZdEI 352 and susceptible line, TZdEI 425 were screened under artificial Striga infestation at Mokwa and Abuja, Nigeria, 2015. Additive‐dominance model was adequate in describing observed variations in the number of emerged Striga plants among the population; hence, digenic epistatic model was adopted for Striga damage. Dominance effects were higher than the additive effects for the number of emerged Striga plants at both locations signifying that non‐additive gene action conditioned inheritance of Striga resistance. Inbred TZdEI 352 could serve as invaluable parent for hybrid development in Striga endemic agro‐ecologies of sub‐Saharan Africa

Explaining Andean Potato Weevils in Relation to Local and Landscape Features: A Facilitated Ecoinformatics Approach

BACKGROUND: Pest impact on an agricultural field is jointly influenced by local and landscape features. Rarely, however, are these features studied together. The present study applies a "facilitated ecoinformatics" approach to jointly screen many local and landscape features of suspected importance to Andean potato weevils (Premnotrypes spp.), the most serious pests of potatoes in the high Andes. METHODOLOGY/PRINCIPAL FINDINGS: We generated a comprehensive list of predictors of weevil damage, including both local and landscape features deemed important by farmers and researchers. To test their importance, we assembled an observational dataset measuring these features across 138 randomly-selected potato fields in Huancavelica, Peru. Data for local features were generated primarily by participating farmers who were trained to maintain records of their management operations. An information theoretic approach to modeling the data resulted in 131,071 models, the best of which explained 40.2-46.4% of the observed variance in infestations. The best model considering both local and landscape features strongly outperformed the best models considering them in isolation. Multi-model inferences confirmed many, but not all of the expected patterns, and suggested gaps in local knowledge for Andean potato weevils. The most important predictors were the field's perimeter-to-area ratio, the number of nearby potato storage units, the amount of potatoes planted in close proximity to the field, and the number of insecticide treatments made early in the season. CONCLUSIONS/SIGNIFICANCE: Results underscored the need to refine the timing of insecticide applications and to explore adjustments in potato hilling as potential control tactics for Andean weevils. We believe our study illustrates the potential of ecoinformatics research to help streamline IPM learning in agricultural learning collaboratives