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

Hongjian Li

TL;DR

This work extends the Fundamental Theorem of Algebra to polynomial matrix equations whose coefficient and unknown matrices lie in the commutative algebra of $Q$-circulant matrices $C(Q)$. By diagonalizing $Q$ through a matrix $T$ with $T^{-1}QT=\mathrm{diag}(\lambda_1,...,\lambda_d)$ and using representation polynomials, the authors reduce the matrix equation to $d$ independent scalar equations $g_i(u_i)=0$, yielding all solutions as $X=T\mathrm{diag}(u_1,...,u_d)T^{-1}$. The polynomials $g_i$ are built from the representation polynomials $f_k$ of the coefficients via $g_i(x)=x^n+f_1(\lambda_i)x^{n-1}+\cdots+f_n(\lambda_i)$. The paper provides corollaries for special choices of $Q$, including $Q(k_1,...,k_d)$ (recovering the circulant case when $k_i=1$) and $C(\pi)$ (companion matrices), and includes explicit examples. Overall, it generalizes Abramov's circulant-result to the broader $Q$-circulant setting, enriching the theory of polynomial matrix equations in structured matrix algebras.

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.

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

TL;DR

This work extends the Fundamental Theorem of Algebra to polynomial matrix equations whose coefficient and unknown matrices lie in the commutative algebra of -circulant matrices . By diagonalizing through a matrix with and using representation polynomials, the authors reduce the matrix equation to independent scalar equations , yielding all solutions as . The polynomials are built from the representation polynomials of the coefficients via . The paper provides corollaries for special choices of , including (recovering the circulant case when ) and (companion matrices), and includes explicit examples. Overall, it generalizes Abramov's circulant-result to the broader -circulant setting, enriching the theory of polynomial matrix equations in structured matrix algebras.

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 -circulant matrices. This result generalizes Abramov's result for circulant matrices.
Paper Structure (6 sections, 9 theorems, 34 equations)

This paper contains 6 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Some Corollaries and Examples
  4. All solutions of equation \ref{['e1']} over $C(Q(k_1,k_2,\dots,k_d))$
  5. All solutions of equation \ref{['e1']} over $C(\pi)$
  6. Conclusion

Key Result

Lemma 1.1

(Li) Let $Q\in M_d(\mathbb{C})$ be a non-derogatory matrix. Then $C(Q)$ is a commutative $\mathbb{C}$-algebra, and $\left\{I,\,Q,\,Q^2,\,\dots,\,Q^{d-1}\right\}$ is a $\mathbb{C}$-basis of $C(Q)$.

Theorems & Definitions (12)

  • Lemma 1.1
  • Definition 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Example 3.1
  • ...and 2 more