Some applications of the comrade matrix

The comrade matrix A is u generalization of the companion matrix associated with a monic polynomial ā(s), and arises when ā(s) is expressed as a linear combination of a set of orthogonal polynomials {pi(s)}. A number of applications which are intensions of results involving the companion form are described. A determinant whose columns are characteristic vectors of A is shown to be a multiple of the Vandermonde determinant. Controllable realizations involving A are constructed, and for a single-input system linear feedback is given which produces an arbitrary closed-loop characteristic polynomial expressed in terms of the pi(s). Methods involving A for locating the zeroes of ā(s) are also discussed. Further possible uses of the comrade matrix are suggested.