\noindent This ongoing work on Homotopic Deviation focuses on the purely algebraic aspect of the theory. We use the definition $\ds (zI - A)^{-1} = \frac{1}{\pi(z)} adj(zI - A) $ for $z \in re(A)=\C \backslash \sigma(A)$, $\pi(z)=$det$(zI-A)$, ~to ~relate $ M_z = V^H (zI - A)^{-1} U $ ~to ~the ~matrix ~polynomial $Q(z) = V^H adj (zI - A ) U$ of order $ r
