For each $m\in\{1, 2, ..., n\}$, is there a transformation $\phi_m$ that I can apply to a matrix $M\in \mathbb{R}^{n\times n}$ such that the $m^{th}$ largest eigenvalue $\lambda_m$ of $M$ is the smallest eigenvalue $\lambda’_n$ of the matrix $M’ = \phi_m (M)$?
Edited for generality.