Polynomials with respect to a general basis. I. Theory

AbstractAs n × n Hessenberg matrix A is defined whose characteristic polynomial is relative to an arbitrary basis. This generalizes the companion, colleague, and comrade matrices when the bases are, respectively, power, Chebyshev, and orthogonal, so the term “confederate” matrix is suggested. Some properties of A are derived, including an algorithm for computing powers of A. A scheme is given for inverting the transformation matrix between the arbitrary and power bases. A Vandermonde-type matrix associated with A and a block confederate matrix are defined