Linear operators on matrices: The invariance of rank-k matrices (Q5925257)

The authors present the following: A characterization is given of all nonsingular linear operators, on the set of \(m\times n\) matrices over any field with at least four elements, which map the set of rank-k matrices into itself. It is shown that if \({\mathcal S}\) is any subspace of \(m\times n\) matrices over any field with at least \(k+1\) elements whose nonzero elements all have rank k, then the dimension of \({\mathcal S}\) is at most max(m,n). This result is used to characterize all linear operators on the set of \(m\times n\) matrices over certain fields which map the set of rank-k matrices into itself.