Inverse Problems for Polynomial and Rational Matrices
Date of Award
Doctor of Philosophy
Dr. D. Steven Mackey
Dr. Niloufer Mackey
Dr. Dennis Pence
Dr. Froilán Dopico
Polynomial matrices, rational matrices, eigenvalues, matrix analysis, linear algebra, inverse problems
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.
Hollister, Richard Allen, "Inverse Problems for Polynomial and Rational Matrices" (2020). Dissertations. 3645.