A note on similarity to contraction for stable $2 \times 2$ companion matrices

Abstract

We consider companion matrices of size $2 \times 2$ with general complex spectra satisfying a root condition with respect to the closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis transformations are constructed such that the resulting, transformed matrix is contractive or dissipative, respectively, with respect to the $\ell_2$-norm.