Numerical “helper solver” to solve polynomial matrix equation system?

I have noticed that when solving the following matrix-polynomial:

$$\sum_{k=0}^N{\bf C}_k{\bf T}^k = {\bf 0} \hspace{0.6cm} \text{ s.t. } \hspace{0.6cm} {\bf C}_k,{\bf T} \in\mathbb{R}^{{M\times M}}$$

using a numerical technique I can expand on if you wish, this solution is sometimes rather difficult to come by. But say that I can solve the equation system where multiply ${\bf C}_k$ with scalars $s_k$ and then manage to find a solution to this system:

$$\sum_{k=0}^{N}s_k{\bf C}_k{\bf T}^k={\bf 0} \hspace{0.6cm} \text{ s.t. } \hspace{0.6cm} {\bf C}_k,{\bf T} \in\mathbb{R}^{{M\times M}}, s_k \in \mathbb{R}$$

Is there any way I could the knowledge of such a solution in the pursuit of a solution to the first one..?