Characterization of matrices having rank k

Let F be a field with at least k+2 elements and let A be an n×n matrix over F. Then rank A≤k if and only if for every invertible matrix B and every distinct nonzero α1,…αk+1∊F at least on of the matrices B+α1 A,…,B+αk+1 A is invertible. This extends the results on spectral characterization of matrices having rank 1 due to Deguang-Limaye [2] as well as the finite-dimensional versions of the results due to Jafarian-Sourour [5] and Limaye [6]. Our proof is much shorter. Using a similar idea we obtain also a characterization of rank one nilpotents among all nilpotents, thus simplifying the proof of the characterization of linear mappings preserving nilpotent matrices.