On the Similarity to Nonnegative and Metzler Hessenberg Forms

We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions $n \geq 3$, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of $n=3$ also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if $n \leq 4$. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions $n \geq 5$ remain similar to Metzler Hessenberg matrices.