Automorphism group of a graph related to zero-divisor graphs

In 2016, the authors of Wang (2016) have determined the automorphisms of the zero-divisor graph $\Gamma \left(M_n\left(F_q\right)\right)$, where the graph is defined in such a way: the vertices are all nonzero zero-divisors of $M_n\left(F_q\right)$ and there is a directed edge from $A$ to $B$ if and only if $AB=0$. Regretfully, the graph $\Gamma \left(M_n\left(F_q\right)\right)$ is over square matrices, directed, finite and with some loops. Thus we are motivated to improve the definition such that the new defined graph can be infinite, undirected, without loops and it can be over rectangular matrices. Let $M_{m,n}\left(F\right)$ be the set of all $m\times n$ rectangular matrices over the field of complex (or real) number. By ${\bar{A}}^T$ we mean the conjugate transpose of $A\in M_{m,n}\left(F\right)$. Let $\Omega \left(M_{m,n}\left(F\right)\right)$ be the graph whose vertex set consists of all nonzero matrices in $M_{m,n}\left(F\right)$ with rank less than $n$, and two vertices $A,B$ are adjacent if and only if $A{\bar{B}}^T=0$ (if $F$ is the real field, that is $A{B}^T=0$). Note that $A{\bar{B}}^T=0$ if and only if $B{\bar{A}}^T=0$ and $A{\bar{A}}^T\ne 0$ for a nonzero matrix $A\in M_{m,n}\left(F\right)$. Thus $\Omega \left(M_{m,n}\left(F\right)\right)$ is infinite, undirected and without loops. In the present paper, the automorphisms of $\Omega \left(M_{m,n}\left(F\right)\right)$ are determined completely.