Hermitian and unitary almost-companion matrices of polynomials on demand

TL;DR

This paper introduces Almost-Companion Matrices (ACMs), a flexible generalization of Companion Matrices, enabling the construction of Hermitian and Unitary matrices for polynomials, with applications in quantum physics and polynomial root-finding.

Contribution

It defines ACMs, demonstrates their construction for specific polynomials, and explores their properties, including conditions for unitarity, extending the companion matrix concept.

Findings

Hermitian and Unitary ACMs can be constructed from third-degree polynomials.

ACMs facilitate solving cubic equations without Cardano-Dal Ferro formulas.

Conditions for polynomial coefficients to correspond to Unitary ACMs are established.

Abstract

We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and Unitary ACMs starting from appropriate third degree polynomials, with implications for their use in physical-mathematical problems such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degree.