J. Rafael Sendra

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We show how to construct linearizations of matrix polynomials $z\mathbf{a}(z)\mathbf{d}_0 + \mathbf{c}_0$, $\mathbf{a}(z)\mathbf{b}(z)$, $\mathbf{a}(z) + \mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right) < \mathrm{deg}\left(\mathbf{a}(z)\right)$), and $z\mathbf{a}(z)\mathbf{d}_0\mathbf{b}(z) + \mathbf{c_0}$ from linearizations of the component parts, $\mathbf{a}(z)$ and $\mathbf{b}(z)$. This allows the extension to matrix polynomials of a new companion matrix construction.

Hoon Hong, Daniel Profili, J. Rafael Sendra

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs.

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $\pm1$. Many properties remain after these specializations, some of which surprised us. We find two recursive formulae for the characteristic polynomials of upper Hessenberg matrices. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix. We count stable matrices, normal matrices, and neutral matrices, and tabulate the results of our experiments. We prove a theorem about the only possible kinds of normal matrices amongst a specific family of Bohemian upper Hessenberg matrices.

Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega et al.

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many properties remain after these specializations, some of which surprised us. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix.