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Dissertation
Constructive solutions to A. Horn's problem, determinantal representations, and real Fejér-Riesz factorization
Doctor of Philosophy (Ph.D.), Drexel University
May 2025
DOI:
https://doi.org/10.17918/00011003
Abstract
In this thesis, we give constructive approaches to a variety of related linear algebra problems. These problems, which include Alfred Horn's problem, real Fejér-Riesz factorization, and determinantal representations of multivariate polynomials, have previously established existence results, but those proofs do not provide a way to construct a solution. Alfred Horn's problem is, given three sets of n real eigenvalues, when do there exist n x n real symmetric matrices A and B so that A has the first set of eigenvalues, B has the second set of eigenvalues, and A+B has the last set of eigenvalues? We now know when A and B exist, but the question remains how do we find A and B? In this thesis, we give ways to find a real Horn solution pair (A,B) in two specific cases. In particular, we first give a construction for A and B when n=3 that uses semidefinite programming. Second, we give a construction for A and B when B is a rank 2 matrix. This construction stems from performing two orthogonal rank 1 updates of A. Our construction of a real Horn solution pair using semidefinite programming when n=3 involves factorizing a positive semidefinite matrix polynomial as a matrix polynomial times its transpose. This is a real version of the Fejér-Riesz factorization, which is known to be possible when the determinant of the positive semidefinite matrix polynomial is the square of a nonzero polynomial. We provide a new, constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation. Our construction of a real Horn solution pair when B is rank 2 highlights a relationship between the Horn problem and the following similar problem: given a set of n real eigenvalues, a set of m real eigenvalues, and a set of n+m real eigenvalues satisfying certain conditions, find an (n+m)x(n+m) real symmetric matrix such that the top left principal submatrix has the first set of eigenvalues, the bottom right principal submatrix has the second set of eigenvalues, and the full matrix has the last set of eigenvalues. This relationship between an additive problem and a block problem leads us to study the interplay between additive and block Hermitian determinantal representations for multivariate polynomials. For each kind of Hermitian determinantal representation, we show necessary and sufficient conditions for a multivariate polynomial to have that representation. Furthermore, we show how to construct such a representation.
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Details
- Title
- Constructive solutions to A. Horn's problem, determinantal representations, and real Fejér-Riesz factorization
- Creators
- Sarah Kathryn Gift
- Contributors
- Hugo J. Woerdeman (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- ix, 159 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991022058837704721