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We introduce the concept of Almost-Companion Matrix (ACM) by relaxing thenon-derogatory property of the standard Companion Matrix (CM). That is, wedefine an ACM as a matrix whose characteristic polynomial coincides with agiven monic and generally complex polynomial. The greater flexibility inherentin the ACM concept, compared to CM, allows the construction of ACMs that haveconvenient matrix structures satisfying desired additional conditions,compatibly with specific properties of the polynomial coefficients. Wedemonstrate the construction of Hermitian and Unitary ACMs starting fromappropriate third degree polynomials, with implications for their use inphysical-mathematical problems such as the parameterization of the Hamiltonian,density, or evolution matrix of a qutrit. We show that the ACM provides a meansof identifying properties of a given polynomial and finding its roots. Forexample, we describe the ACM-based solution of cubic complex algebraicequations without resorting to the use of the Cardano-Dal Ferro formulas. Wealso show the necessary and sufficient conditions on the coefficients of apolynomial for it to represent the characteristic polynomial of a unitary ACM.The presented approach can be generalized to complex polynomials of higherdegree.

Hermitian and unitary almost-companion matrices of polynomials on demand