Linear operators on matrices: the invariance of rank-k matrices

AbstractA characterization is given of all nonsingular linear operators, on the set of m×n matrices over any field with at least four elements, which map the set of rank-k matrices into itself. It is also shown that if L is any subspace ofm×n matrices over any field with at least k+1 elements whose nonzero elements all have rank k, then the dimension of L is at most max(m,n). This fact is used to characterize all linear operators on the set of m×n matrices over certain fields which map the set of rank-k matrices into itself