A new kind of companion matrix

Abstract

A new kind of companion matrix is introduced, for polynomials of the form c(λ) = λa(λ)b(λ)+c_0, where upper Hessenberg companions are known for the polynomials a(λ) and b(λ). This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrenceâs Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials, which are also new to this paper.