For a given scalar monic polynomial of the form $p: \mathbb{C} \rightarrow \mathbb{C}, p(x) = c_0 + c_1 x + \cdots + x^n$, one can compute the roots of this polynomial by computing the eigenvalues of its companion matrix.
According to various sources, this matrix can take multiple forms such as $$ A[p]=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \end{bmatrix}.$$
If one considers now a matrix valued polynomial of the form $$P: \mathbb{C} \rightarrow \mathbb{C}^{k\times k}, P(x) = \mathbf{C}_0 + \mathbf{C}_1 x + \cdots + \mathbf{C}_n x^n$$ what should be the associated companion matrix ? Note the presence of the $\mathbf{C}_n$ coefficient, that may not be invertible.