Extension of the Fundamental Theorem of Algebra to Polynomial Matrix Equations over $Q$-Circulant Matrices

TL;DR

This paper extends the Fundamental Theorem of Algebra to polynomial matrix equations involving $Q$-circulant matrices, broadening the scope of algebraic solutions for structured matrix equations.

Contribution

It generalizes Abramov's result for circulant matrices to the broader class of $Q$-circulant matrices, establishing an algebraic foundation for these matrix equations.

Findings

Established an analogue of the Fundamental Theorem of Algebra for $Q$-circulant matrices.

Generalized previous results from circulant to $Q$-circulant matrices.

Provided theoretical groundwork for solving polynomial matrix equations with $Q$-circulant structure.

Abstract

In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result for circulant matrices.