## Matrix Polynomials

Since its inception, the theory of matrix polynomials has been strongly influenced by its applications to differential equations and vibrating systems. In fact, vibrating systems 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. 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 spectra, and it is in this spirit that we have focused on similar generalizations from matrix theory and complex analysis.

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 with inner radius 1/2 and outer radius 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 a priori 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 self-adjoint matrix polynomials whose coefficients are either positive or negative definite, or null. Our main result shows that this generalization holds under the additional assumption that the matrix polynomial is hyperbolic. Also, we prove special cases where the matrix polynomial is diagonalizable by congruence, or of degree three or less. We conjecture that this generalization holds for all self-adjoint matrix polynomials whose coefficients are either positive or negative definite, or null, and discuss what makes this open problem non-trivial.

In~\cite{Cameron2019_GH}, we present a generalization of Householder sets for matrix polynomials. After defining these sets, we analyze their topological and algebraic properties, which include containing all of the eigenvalues of a given matrix polynomial. Finally, we show that these Householder sets are intimately connected with other inclusion sets, such as the Ger{\v s}gorin set and pseudospectra of a matrix polynomial, as well as with the Bauer-Fike theorem.