Narayana, Mandelbrot, and A New Kind of Companion Matrix

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