Let $A$ be a $n \times n$ matrix. We know that $\lambda$ is an eigenvalue of $A$ and $x$ is its associated eigenvector if together they satisfy $Ax = \lambda x$. We can also write this equation as $Ax = \lambda I_n x$, where $I_n$ is the $n \times n$ identity matrix. If we want to be even more unnecessary, we can write this equation as
\begin{align} (A \otimes I_k) (x \otimes I_k) &= (\lambda I_n \otimes I_k) (x \otimes I_k) \\ &= (I_n \otimes \lambda I_k) (x \otimes I_k) \end{align}
where $``\otimes"$ denotes the kronecker product. Recall that $(A \otimes B) (C \otimes D) = (AB \otimes CD)$ when the products $AB$ and $CD$ make sense.
I am interested in the following generalization: let $B$ be $nk \times nk$. What can we say about matrices $X$ of dimension $nk \times k$ and $\Lambda$ of dimension $k \times k$ that satisfy
\begin{align} B X = (I_n \otimes \Lambda) X \end{align}
Evidently, $k = 1$ degenerates to the regular eigenvalue problem. I am not sure if the problem is interesting as is, I am open to suggestions for different setups.