When R is a commutative ring, matrices A and B in Mn(R) are called conjugate when UAU−1 = B for some U ∈ GLn(R). The conjugacy problem in Mn(R) is: decide when two matrices in Mn(R) are conjugate. We want to look at the conjugacy problem in Mn(Z), where ideal theory and class groups make an interesting appearance. The most basic invariant for conjugacy classes of matrices is the characteristic ...

In this paper, analytic expressions are derived for the complex and real stability radii of non-monic polynomial matrices with respect to an arbitrary stability region of the complex plane. Numerical issues for computing these radii for different perturbation structures are also considered with application to robust stability of Hurwitz and Schur polynomial matrices.

Some block Toeplitz methods applied to polynomial matrices are reviewed. We focus on the computation of the structure (rank, null-space, infinite and finite structures) of an arbitrary rectangular polynomial matrix. We also introduce some applications of this structural information in control theory. All the methods outlined here are based on the computation of the null-spaces of suitable block...

We consider polynomial maps described by so-called (multivariate) linearized polynomials. These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most b...

This paper addresses robust D-stability analysis problems of uncertain polynomial matrices. The underlying idea we follow is that a given polynomial matrix is D-stable if and only if there exist polynomial-type multipliers that render the resulting polynomial matrices to be strictly positive over a specific region on the complex plane. By applying the generalized S-procedure technique, we show ...

Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic po...

Laplacian pyramid based Laurent polynomial (LP2) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in Signal Processing. In this paper, we investigate when such matrices are scalable, that is when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scala...

Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding polynomial. In the classical companion pencil case (A,B) only the coefficient of the highest degree appears in B’s lower right corner. We will show, however, that all coefficients of the polynomial ...

Abstract: The universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems, and proved in some particular cases, as the hermitian one-matrix model with polynomial potential. Here, we consider the more difficult case of a unidimensional chain of matrices with first neighbour couplings and polynomial potentials. An asymptotic expressi...

In this short paper we study the existence of positive and negative semide nite solutions of the algebraic Riccati equation corresponding to linear quadratic problems with an inde nite cost functional. An important role is played by certain two-variable polynomial matrices associated with the algebraic Riccati equation. We characterize al unmixed solutions in terms of the Pick matrices associat...