Congenial matrices

For a given polynomial in the usual power form, its associated companion matrix can be applied to investigate qualitative properties, such as the location of the roots of the polynomial relative to regions of the complex plane, or to determine the greatest common divisor of a set of polynomials. If the polynomial is in “generalized” form, i.e. expressed relative to an orthogonal basis, then an analogue to the companion matrix has been termed the comrade form. This followed a special case when the basis is Chebyshev, for which the term colleague matrix had been introduced. When a yet more general basis is used, the corresponding matrix has been named confederate. These constitute the class of congenial matrices, which allow polynomials to be studied relative to an appropriate basis. Block-partitioned forms relate to polynomial matrices.