A practical method is presented for determining root and pole cancellation functions of a matrix function $Q(z)$ meromorphic on the extended complex plane $\bar{ \mathbb{ C}}:=\mathbb{ C} \cup \left\{ \infty \right\}$. This method is applied to solve a nonlinear system of $n\in \mathbb{ N}$ differential equations of order $l\in \mathbb{ N}$ with $n $ unknown functions $u_{ i}\left( t \right)$, where $i=1,\, \mathellipsis ,\,n $. For a function $Q\in \mathcal{ N}_{ \kappa}(\mathcal{ H}) ,\, \kappa \in \mathbb{ N} \cup \lbrace 0 \rbrace$, posesing a pole at infinity of order $m \in \mathbb{ N}$, the following factorization is establish \[ Q(z)=(z-\beta)^{ m}\tilde{ Q}(z), \, z\in \mathcal{ D}(Q), \] where $\beta \in \mathbb{ R}$ is a regular point of $Q$, and $\tilde{ Q}\in \mathcal{ N}_{ \kappa'}(\mathcal{ H})$ is holomotphic at $\infty$. Unlike the Krein-Langer representation of $Q$, which involves a linear relation $A$, this representation employs a bounded operator $\tilde{ A}$ in the Krein-Langer representation of $\tilde{ Q}$. The operator $\tilde{ A}$ and the relation $A$ have identical spectra, except at $\beta$ and $\infty$. We demonstrate how to obtain this representation for a given meromorphic function $Q\in \mathcal{ N}_{ \kappa}^{ n \times n}$ using the root functions developed in this work.
Categories: math.CV math.FA