$\newcommand\la\lambda\newcommand\La\Lambda$
>Given a square matrix $M$ of size $n\times n$, find a matrix $V$ of size $n\times 2$ and a matrix $\Lambda$ of size $2\times 2$ such that
$$
MV = V\Lambda.
$$

There may be many such matrices, even with the additional condition that $\La$ be diagonal. E.g., if $u$ and $w$ are eigenvectors of $M$ with the corresponding eigenvalues $\la$ and $\mu$, $V$ is the $n\times2$ matrix with columns $u$ and $w$, and $\La$ is the diagonal matrix with $\la$ and $\mu$ on the diagonal, then $MV=V\La$ will hold.