The study of the projective transformation under the bilinear strict equivalence

The study of linear time invariant descriptor systems has intimately been related to the study of matrix pencils. It is true that a large number of systems can be reduced to the study of differential (or difference) systems, $S\left ( {F,G} \right )$, $$\begin{align*} & S\left({F,G}\right): F\dot{x}(t) = G{x}(t) \left(\text{or the dual, } F{x}(t) = G\dot{x}(t)\right), \end{align*}$$and $$\begin{align*} & S\left({F,G}\right): Fx_{k+1} = Gx_k \left(\text{or the dual, } Fx_k=Gx_{k+1}\right)\!, F,G \in{\mathbb{C}^{m \times n}}, \end{align*}$$and their properties can be characterized by homogeneous matrix pencils, $sF - \hat{s}G$. Based on the fact that the study of the invariants for the projective equivalence class can be reduced to the study of the invariants of the matrices of set ${\mathbb{C}^{k \times 2}}$ (for $k \geqslant 3$ with all $2\times 2$-minors non-zero) under the extended Hermite equivalence, in the context of the bilinear strict equivalence relation, a novel projective transformation is analytically derived.