Print Email Facebook Twitter Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand Title Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand Author Markovich, L. (TU Delft QID/Borregaard Group; TU Delft QuTech Advanced Research Centre; Universiteit Leiden; Kavli institute of nanoscience Delft; National University of Science and Technology MISiS; Russian Quantum Center, Skolkovo; Institute for Information Transmission Problems, Moscow) Migliore, Agostino (Università degli Studi di Padova) Messina, Antonino (Università degli Studi di Palermo) Date 2023 Abstract We introduce the concept of the 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 the 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 degrees. Subject almost-companion matrixcompanion matrixcomplex polynomialdensity matrixhermitian matrixsub-parameterizationunitary matrix To reference this document use: http://resolver.tudelft.nl/uuid:e606d63d-8121-43ba-b806-02c6583708cb DOI https://doi.org/10.3390/e25020309 ISSN 1099-4300 Source Entropy: international and interdisciplinary journal of entropy and information studies, 25 (2) Part of collection Institutional Repository Document type journal article Rights © 2023 L. Markovich, Agostino Migliore, Antonino Messina Files PDF entropy_25_00309.pdf 505.51 KB Close viewer /islandora/object/uuid:e606d63d-8121-43ba-b806-02c6583708cb/datastream/OBJ/view