A companion matrix analogue for orthogonal polynomials

Abstract

Given a set of orthogonal polynomials {Pi(x)}, it is shown that associated with a polynomial a(x)=∑aipi(x) there is a matrix A which possesses several of the properties of the usual companion form matrix C. An alternative and possibly preferable form A' is also suggested. A similarity transformation between A [orA'] and C is given. If b(x) is another polynomial then the matrix b(A) [or b(A')] has properties like those of b(C), relating to the greatest common divisor of a(x) and b(x).