On preservers of singularity and nonsingularity of matrices

Abstract

Operators preserving singularity and nonsingularity of matrices were studied in paper of P. Botta under the assumption that operators are linear. In the present paper the linearity of operators is not assumed: we only assume that operators are of the form T = (fi,j), where f i j : K —• K and K is a field, i,j € {1,2, . . . , n } . If n > 3, then in the matrix space Mn(K) operators preserving singularity and nonsingularity of matrices must be as in paper of P. Botta. If n < 2, operators may be nonlinear. In this case the forms of the operators are presented. Let R, N denote the set of real numbers or positive integer numbers, respectively. Let Mn(K) be the set of n x n matrices over a field K, i.e. Mn{K) e Knxn, where n e N. We denote by Ej^ the matrix whose j,k entry is 1 and the remaining entries of which are 0. First of all let us introduce.