Linear bijective maps preserving invertibility on pairs of similar matrices

Abstract

Let $n\ge 2$ be a natural number, and denote by $M_n$ the space of all $n\times n$ matrices over the complex field. In this paper, we characterize linear bijective maps φ on $M_n$ having the property that if $A,B\in M_n$ are similar matrices and $\phi \left(A\right)$ is invertible, then $\phi \left(B\right)$ is invertible as well.