The automorphism group and fixing number of orthogonality graph over a vector space

Let F be a field, and F-n the n-dimensional row vector space over F. The orthogonality graph O(F-n) of F-n is an undirected simple graph which has (F-n)* = F-n\{0} as its vertex set, and for distinct alpha, beta is an element of (F-n)*, alpha similar to beta if and only if alpha beta(T) = 0, where beta(T) is the transpose of beta. When F is finite, it, is shown that, any automorphism of O(F-n) can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of O(F-n) are considered.