On affine homogeneous polynomial centrosymmetric matrices
Miriam Manoel, Leandro Nery
TL;DR
The paper introduces DNA matrices, a new polynomial-entry class of centrosymmetric matrices tied to invariant polynomials on the Minkowski plane under hyperbolic (Lorentz) rotations. It proves these matrices are centrosymmetric for arbitrary $\alpha$ and $\beta$, and shows that when $\alpha^2-\beta^2=1$ the singularity of the matrices is governed by parity: odd-order matrices are nonsingular while even-order ones have a 1-dimensional nullspace, with an explicit nullvector pattern. Using Hilbert basis methods for hyperbolic invariants, the authors establish that the invariant ring is generated by $||(x,y)||_1^2 = x^2-y^2$, and they derive the nullspace description accordingly. The work provides a concrete bridge between invariant theory and matrix structure, yielding explicit computations, a clear parity-based theory, and potential avenues for generalizations to higher dimensions or other group actions.
Abstract
We explore a class of centrosymmetric matrices whose entries are polynomials in two variables, referred to as DNA matrices. Our motivation stems from an unexpected connection between these matrices and invariant polynomials under the action of a Lorentz rotation on the plane. Among several noteworthy properties, we establish that within a subclass of DNA matrices, singular matrices occur precisely when their order is even, and we determine their null space in such cases. The results provide new insights into the structural characteristics of centrosymmetric matrices, expanding their theoretical foundations and potential applications.