## Matrix Polynomials

Since its inception, the theory of matrix polynomials has been influenced by its applications to differential equations and vibrating systems. In fact, vibrating systems strongly motivated the first works devoted primarily to matrix polynomials: one by Frazer, Duncan, and Collar in 1938~\cite{Frazer1955} and the other by Lancaster in 1966~\cite{Lancaster1966}. In addition, the theory of matrix polynomials has been influenced by results from matrix theory and complex analysis. For instance, the canonical set of Jordan chains defined in~\cite{Gohberg2009} are a natural generalization of a cycle of generalized eigenvectors, which are defined in~\cite{Friedberg2003}. A generalized Rouch\'e's theorem is presented in~\cite{Gohberg1990} and this result is used to prove a generalized Pellet's theorem for matrix polynomials~\cite{Melman2013}. These generalizations have benefited our study of matrix polynomials and their spectrum, and it is in this spirit that we have focused on similar generalizations from the theory of matrices and polynomials.

In~\cite{Cameron2015}, we use the generalized Rouch\'e theorem to provide spectral bounds for unitary matrix polynomials. This result can be viewed as a generalization of the fact that the eigenvalues of a unitary matrix lie on the unit circle or the fact that a scalar polynomial with unimodular coefficients has roots that lie in the annulus $\mathcal{A}(1/2,2)=\{z\in\mathbb{C}\colon~\frac{1}{2}<|z|<2\}$.

In~\cite{Cameron2016}, we use a pseudo inner product to provide a constructive proof that every matrix polynomial can be reduced to a matrix rational in Hessenberg form. In general, an apriori reduction of a matrix polynomial to a simpler form can be used to reduce the cost of computing its eigenvalues. For additional methods capable of reducing a matrix polynomial to a simpler form, see~\cite{Nakatsukasa2018}.

In~\cite{Cameron2018_DRSMP}, we present a generalization of Descartes' rule of signs for matrix polynomials. This result provides an upper bound on the number of real positive eigenvalues of a self-adjoint matrix polynomial whose coefficients are positive/negative definite, or otherwise null. Furthermore, this result has applications to overdamped vibration systems, which are discussed in~\cite{Duffin1955,Gohberg2009,Lancaster1966,Tisseur2001}.

### Student Research

Currently, I am working with Davidson College student Michael Robertson on inclusion sets for the eigenvalues of a matrix polynomial. We are studying which inclusion sets give more information about the spectral set of a matrix polynomial. Applications for this work include initial estimates for iterative methods that compute the eigenvalues of a matrix polynomial.

Previous work includes a project with College of Idaho student Leo Trujillo on the numerical range of a matrix polynomial. Our work consisted of writing Python code to create plots of the numerical range and test a conjecture on Descartes' rule of signs for matrix polynomials.