A matrix-polynomial structure in a finite-dimensional vector space

The classical interpretation of a matrix is the representation of an operator in a fixed coordinate system. Another interpretation of a symmetric matrix is the representation of a quadratic form. In this article we propose a new concept by considering the following three objects simultaneously: (i) a strongly nonsingular matrix, (ii) nonfactorable low and upper Hessenberg matrices, and (iii) two systems of special polynomials.