Polynomial matrices with given determinant

Counting Integral Matrices with a given Characteristic Polynomial

We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices, in large balls, with a given monic integral irreducible polynomial as their common characteristic polynomial. The proof uses equidistributions of polynomial trajectories on SL(n, R)/SL(n, Z), which is a generalization of Ratner’s theorem on equidistributions of unipotent trajectories....

Sampling Real Matrices with Given Margins

Extension of matrices with Laurent polynomial entries

Let P and Q be nr, r n matrices with Laurent polynomial entries. Suppose that Q T (z)P (z) = I r , z 2 Cnf0g. This note provides an algorithmatic construction of two n(nr) matrices G and H with Laurent polynomial entries such that the nn matrices

On the determinant of quaternionic polynomial matrices and its application to system stability

The research reported in this paper is motivated by the study of stability for linear dynamical systems with quaternionic coefficients. These systems can be used to model several physical phenomena, for instance, in areas such as robotics and quantum mechanics. More concretely, quaternions are a powerful tool in the description of rotations [1]. There are situations, especially in robotics, whe...

Quaternionic Matrices: Inversion and Determinant ∗

We discuss the Schur complement formula for quaternionic matrices, M , and give an efficient method to calculate the matrix inverse. We also introduce the functional D[M ] which extends to quaternionic matrices the non-negative number |det[M ]|. 1. Introduction. Much of the spectral theory of complex matrices does not extend to quaternion matrices without further modifications [1, 2, 3]. In par...