Inverse Problems For Polynomial And Rational Matrices
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Inverse Problems for Polynomial and Rational Matrices
Inverse problems have long been studied in mathematics not only because there are many applications in science and engineering, but also because they yield new insight into the beauty of mathematics. Central to the subject of linear algebra is the eigenvalue problem: given a matrix, and its eigenvalues (numerical invariants). Eigenvalue problems play a key role in almost every field of scientific endeavor from calculating the vibrational modes of a molecule to modeling the spread of an infectious disease, and so have been studied extensively since the time of Euler in the 18th century. If a typical matrix eigenvalue problem asks for the eigenvalues of a given matrix, an inverse eigenvalue problem asks for a matrix whose eigenvalues are a given list of numbers. For matrices over an algebraically closed field, the inverse eigenvalue problem is completely and transparently solved by the Jordan canonical form. If the field is not algebraically closed, there are similar, albeit more involved, solutions, a prime example of which is the real Jordan form when the field is the real numbers. Eigenvalue and inverse eigenvalue problems go beyond just matrices with fixed scalar entries. They have been studied for matrix pencils, which are matrices whose entries are degree-one polynomials with coefficients from a field. A polynomial matrix is a matrix whose entries are polynomials with coefficients from a field. The story of eigenvalues for polynomial matrices (of which matrix pencils are a special case) is more complicated because of the possibility of an infinite eigenvalue. In addition, for singular polynomial matrices, there are invariants that characterize the left and right null spaces called minimal indices. The collection of all this data (finite and infinite eigenvalues together with minimal indices) is known as the structural data of the polynomial matrix. In this dissertation, the inverse structural data problem for polynomial matrices is considered and solved. We begin with the history of this inverse problem, including known results and applications from the literature. Then a new solution is given that is sparse and transparently reveals the structural data in much the same way that the Jordan canonical form transparently reveals the structural data of a scalar matrix. The dissertation concludes by discussing the inverse problem for rational matrices (matrices whose entries are rational functions over a field) and presenting a solution adapted from the solution for the polynomial matrix inverse problem.
Matrix Polynomials
This book is the definitive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener-Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials. The book is appropriate for students, instructors, and researchers in linear algebra, operator theory, differential equations, systems theory, and numerical analysis. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex analysis.
Orthogonal Matrix-valued Polynomials and Applications
This paper is a largely expository account of the theory of p x p matrix polyno mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n" ... , hn of p x p matrices with h-i = hj for j = 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, ... , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . '-0 ' I- whenever Hk is invertible.