Automorphism group of the symmetry trace graph of real matrices

Let R be the real field and M-m,M-n(R) be the set of all m x n matrices over R, where m,n >= 2. For a square matrix A is an element of M-m,M-m(R), tau(A) denotes the trace of A (the sum of all diagonal entries of A). The symmetry trace graph Gamma(t)(M-m,M-n(R)) of M-m,M-n(R) is defined to be a graph with vertex set of all nonzero matrices in M-m,M-n(R) and two vertices A and B are adjacent if and only if tau(AB')=0, where B' is the transpose of B. Clearly, Gamma(t)(M-m,M-n(R)) is undirected and without loops. In the present paper, by studying maximum cliques of Gamma(t)(M-m,M-n(R)), we determine the form of an arbitrary automorphism of Gamma(t)(M-m,M-n(R)).