Linear preservers of matrices of rank-2

Abstract

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.

Keywords:

• Preserver
• Rank
• (U-K)-operator
• AMS Classifications: 15A04, 15A33