I have a matrix equation in the form of:
$$ \begin{equation} \begin{bmatrix} 0 & y_{1} & \cdots & y_{k} \\ y_{1} & & & \\ \vdots & & A + t(B + C) + t^2 D & \\ y_{k} & & & \end{bmatrix} \begin{bmatrix} \Delta \mu \\ \Delta \alpha_{1} \\ \cdots \\ \Delta \alpha_{k} \end{bmatrix} = - \begin{bmatrix} 0 \\ (t(B + C) + t^2 D)\boldsymbol{\alpha} \end{bmatrix} \end{equation} $$
where $\Delta \mu$ is a scalar, $\boldsymbol{\alpha}$ and $\Delta \boldsymbol{\alpha}$ are vectors of size $k$, and matrices $A, B, C, D$ are of size $k\times k$.
I am looking to convert this into something of the following form:
$$ \begin{equation} \begin{bmatrix} \Delta \mu \\ \Delta \alpha_{1} \\ \cdots \\ \Delta \alpha_{k} \end{bmatrix} = G \begin{bmatrix} 1 \\ t \\ t^2 \end{bmatrix} \end{equation} $$
I have gone as far as converting the right side of the equation to
$$ - \begin{equation} \begin{bmatrix} 0 & 0 \\ (B + C)\boldsymbol{\alpha} & D\boldsymbol{\alpha} \end{bmatrix} \begin{bmatrix} t \\ t^2 \end{bmatrix} \end{equation} $$ but I haven't gotten much further with the left side. I especially have trouble getting all terms containing $t$ to the right side, while keeping $\Delta \boldsymbol{\alpha}$ on the left side. Is my problem even possible to solve?