On the construction of nonnegative symmetric and normal matrices with prescribed spectral data

Sherod Eubanks

doi:10.7273/000005954

Nonnegative matrices appear in many branches of mathematics, as well as in applications to other disciplines such as economics, computer science, and chemistry. Since the inception of the fundamental results by Perron and Frobenius, the area of nonnegative matrices have been a fertile field for research. In this dissertation, we consider the problem of reconstructing a nonnegative symmetric or normal matrix based on a knowledge of spectral data. Specifically, our exposition is centered around two facets of the just-mentioned problem. First, we consider symmetric matrices with spectrum [sigma] = {[lamba]1, [lamba]2, . . . , [lamba]n} and corresponding orthonormal set of eigenvectors s1, s2, . . . , sn, such that successive spectral decompositions are nonnegative: Xti=1[lamba]isisTi ? 0, t = 1, . . . , k. We determine the zero-nonzero structure of the si's which correspond to positive [lamba]i's, and provide a complete characterization of the si's in case the above holds for [lamba]1 = [lamba]2 = = [lamba]t = 1 for t = 1, . . . , n. The resulting orthogonal matrices S = [s1, s2, . . . , sn], which we call here extended Soules matrices, are then a generalization of the well-studied class of Soules matrices (henceforth called classical Soules matrices). Among other results, we also prove each extended Soules matrix is the limit of a sequence of classical Soules matrices, and that the rank of symmetric matrices whose eigenvectors form an extended Soules matrix is equal to the cp-rank of the matrix. The other associated problem is that of characterizing the set of potential [lamba]i's, where [lamba]1 ? [lamba]2 ? ? [lamba]n ? 0, such that the above partial sum nonnegativity is accomplished for a given fixed set s1, s2, . . . , sn. We provide some initial results in this direction, as well as an example of how such an analysis would proceed using certain orthogonal Hadamard matrices. Second, we consider the nonnegative (nonsymmetric) normal inverse eigenvalue problem (NNIEP), which is the problem of determining necessary and sufficient conditions on a list [sigma] of complex numbers such that [sigma] is the spectrum of a nonnegative normal matrix. We give a summary of some known necessary and sufficient conditions for the NNIEP, and present some preliminary results using the somewhat new technique of analyzing the eigenvectors of certain skew-symmetric matrices, and using the result to construct solution matrices for the NNIEP. Using this technique, we are able to give the strongest possible result for the NNIEP for 3 3 matrices, and make some progress on the NNIEP for 4 4 matrices. That our approach has promise is evidenced by the nonnegative normal matrix we construct whose spectrum is [sigma] = {12?2, ?12, 12 + i, 12 ? i}, even though [sigma] satisfies none of the currently known sufficient conditions for the NNIEP.

On the construction of nonnegative symmetric and normal matrices with prescribed spectral data

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