# Matrix transformation for finding eigenvalues

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.

• In absolute value or not ? Because for $M'=-M$ you get to invert the ordering on eigenvalues... If kernel is empty, you can take $M'=M^{-1}$ it transforms $\lambda$ in $\frac 1\lambda$. But in both these examples we do not have $\lambda_1=\lambda'_n$, just some relation between them. Do you want equality for just this one eigenvalue ? – zwim May 14 at 23:28
• Note that if you know $\lambda_1$ then $M'=2\lambda_1I-M$ works. Maybe if you are able to have an upper bound $m$ for the eigenvalues, then $M'=2mI-M$ would also suits your needs (for an algorithm for instance). – zwim May 14 at 23:40
• Edited post for generality. – Josh Payne May 17 at 7:22