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The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized by a Hessenberg form. We demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.

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.

In this paper, the problem of locating the left eigenvalues of the quaternion matrices and their connection with the zeros of the quaternion polynomials with quaternion coefficients is considered by using various matrix tools. As an application of which, various famous results for locating the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients are obtained, which include the extension of Cauchy’s theorem, Parodi’s theorem as well.

This study introduces a relationship between the Chebyshev polynomial and the matrix-less method. In this way, it explains the roots of the Chebyshev polynomial and their role in the theory of matrix polynomials. With the help of these techniques, we can evaluate the determinant of some band matrices and pay special attention to the polynomial eigenvalue problem for matrix polynomials expressed in the nonmonomial bases. Numerical results are presented and compared with corresponding theoretical data.

In this paper, we apply several matrix inequalities to the generalized companion matrix of monic-polynomial and thereby obtain some new estimates for the moduli of their zeros.

Using a variety of matrix techniques, the problem of locating the left eigenvalues of the quaternion companion matrices is investigated in this paper. In a recent paper, Dar et al. [8], proved that the zeros of a quaternionic polynomial and the left eigenvalues of corresponding companion matrix are same. In view of this, we use various newly developed matrix techniques to prove various results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients, which include an extension of the result of A. L. Cauchy as well.

A short proof is given for the Cayley-Hamilton theorem using the concept of the subminimal polynomial.