Consider the following matrix equation for $A$: $$\left[\underbrace{X^{i}}_{2\times2}\begin{bmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{bmatrix} - \begin{bmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{bmatrix}\underbrace{A^{i}}_{4\times4} \right]\underbrace{Y}_{4\times3}=0\quad \forall i\in\{0,1,2,\dots\}$$ for some given $X,Y$. The dimensions of $A,X,Y$ are indicated underneath the braces.
Note that there always exists a trivial solution: $$A=\begin{pmatrix}X_{11} & X_{12} & 0 & 0\\ X_{21} & X_{22} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.$$
I have two questions:
- Will there always be other solutions, regardless of $Y$?
- Can one characterize the complete set of solutions for $A$ in terms of $X$ and $Y$ without referring to $i$?. E.g., are there necessary and sufficient conditions on $A$ which can be checked using $X$ and $Y$ (but not $i$)?