Dissertation
Minimal realizations and determinantal representations in the indefinite setting
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2020
DOI:
https://doi.org/10.17918/00001087
Abstract
For rational matrix functions in the one variable Schur class finite dimensional contractive realizations exist. That is, if M(z) is a rational matrix function that is analytic and contractive on the unit disk, there exists a contractive block matrix [A B / C D] so that M(z) = D + zC(I - zA)⁻¹B , |z|<1. The theory of realizations is of importance in control and systems theory and in interpolation problems, and it provides a useful tool in operator theory in general. The above realization result was also used in obtaining a determinantal representation result for stable polynomials of two variables. Namely, a two variable polynomial p(z₁,z₂) of degree (n₁, n₂) without roots on the open bidisk can be shown to allow the representation p(z₁,z₂) = p(0,0) det(I_[n₁+n₂] - KZ), Z = z₁I_[n₁] [o plus] z₂I_[n₂] , where K is a contraction. In the case when p(z₁,z₂) does not have roots on the closed bidisk, the matrix K can be chosen to be a strict contraction. In this thesis we develop generalizations of these results where matrices (and operators) are no longer contractive in the traditional sense, but now with respect to an indefinite inner product. Equivalently, the condition I - K^*K being positive (semi)definite is replaced by the condition that J - K^*JK is positive (semi)definite where J = J^* = J⁻¹ is a signature matrix (or operator) representing the indefinite inner product. The results presented in this thesis involve minimal realization results for rational matrix functions that are J-contractive and strictly J-contractive. In addition, it is shown that under certain conditions a two variable polynomial p(z₁,z₂) without roots on the bitorus or at the origin allows a determinantal representation p(z₁,z₂) = p(0,0) det(I_[n₁+n₂] - KZ), Z = z₁I_[n₁] [o plus] z₂I_[n₂] , where (n₁,n₂) is the degree of the polynomial and K is a J-contraction for an appropriate signature matrix J. The negative signature of J is determined by the number of roots of p(z,z) inside the open unit disk. The techniques that are used include reproducing kernel Pontryagin spaces, the Hahn-Banach separating hyperplane theorem, the theory of Bezoutians, and canonical factorization of matrix-valued trigonometric polynomials.
Metrics
44 File views/ downloads
25 Record Views
Details
- Title
- Minimal realizations and determinantal representations in the indefinite setting
- Creators
- Joshua Dunne Jackson
- Contributors
- Hugo J. Woerdeman (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- vii, 78 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991014695235704721