On invertible preservers of singularity and nonsingularity of matrices over a field

Abstract

Invertible operators preserving singularity of matrices were studied in [3] and [4] under assumption that operators are linear. In the present paper the linearity of operators is not assumed: we assume only that operators are of the form F = (fi,j), where fi,j : F −> F and F is a field, i, j {1, 2, . . . , n}. If n ≥ 3, then in the matrix space Mn(F) operators preserving singularity of matrices must be as in [1]. If n ≤ 2, then operators may be nonlinear. In this case the forms of the operators are presented.