Definition (Proper function)

A Rational Function $g$ where $$g(s)=\frac{b_{m}s^{m}+\dots+b_{1}+b_{0}}{s^{n}+a_{n-1}s_{n-1}+\dots+a_{1}s+a_{0}}$$is called proper if $\lim_{ s \to \infty }g(s)$ exists or $n\geq m$. and is called strictly proper if $\lim_{ s \to \infty }g(s)=0$ or $n>m$.

Definition (Set of Rational & Proper Matrix Functions)

Let $RP$ denote the set of matrix functions which are real rational and proper.