Nonlinear mappings preserving at least one eigenvalue

TL;DR

This paper characterizes nonlinear maps on complex matrices that preserve at least one eigenvalue in differences, showing they are essentially conjugations or transpose conjugations, under Lipschitz or differentiability assumptions.

Contribution

It establishes that such eigenvalue-preserving maps must be conjugation or transpose conjugation transformations, extending previous linear-preserving results to nonlinear Lipschitz and differentiable maps.

Findings

Maps are either conjugation or transpose conjugation.

Results hold under Lipschitz or -class differentiability.

Spectral radius-preserving maps are characterized similarly.

Abstract

We prove that if $F$ is a Lipschitz map from the set of all complex $n\times n$ matrices into itself with $F(0)=0$ such that given any $x$ and $y$ we have that $% F\left( x\right) -F\left( y\right) $ and $x-y$ have at least one common eigenvalue, then either $F\left( x\right) =uxu^{-1}$ or $F\left( x\right) =ux^{t}u^{-1}$ for all $x$, for some invertible $n\times n$ matrix $u$. We arrive at the same conclusion by supposing $F$ to be of class $\mathcal{C}% ^{1}$ on a domain in $\mathcal{M}_{n}$ containing the null matrix, instead of Lipschitz. We also prove that if $F$ is of class $\mathcal{C}^{1}$ on a domain containing the null matrix satisfying $F(0)=0$ and $\rho (F\left( x\right) -F\left( y\right) )=\rho (x-y)$ for all $x$ and $y$, where $\rho \left( \cdot \right) $ denotes the spectral radius, then there exists $\gamma \in \mathbb{C}$ of modulus one such that either $\gamma ^{-1}F$ or $\gamma ^{-1}\overline{F}$ is of the above form, where $\overline{F}$ is the (complex) conjugate of $F$.