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We demonstrate a new kind of companion matrix, for polynomials of the form$c(\lambda) = \lambda a(\lambda)b(\lambda) + c_0$ where upper Hessenbergcompanions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. Thisconstruction can generate companion matrices with smaller entries than theFiedler or Frobenius forms. This generalizes Piers Lawrence's Mandelbrotcompanion matrix. We motivate the construction by use of Narayana-Mandelbrotpolynomials, which are also new to this paper.

Narayana, Mandelbrot, and A New Kind of Companion Matrix