For F : M nn M nn defined by F(A) = A T AI, DF(A)H = A T H + H T A. Let O(n) := {A M nn : A T A = I} = {A M nn : F(A) = 0 }, where 0 denotes the n n...

For F : Mn×n → Mn×n defined by F(A) = AT A−I, DF(A)H = AT H + HT A. Let O(n) := {A ∈ Mn×n : A T A = I} = {A ∈ Mn×n : F(A) = 0}, where 0 denotes the n × n matrix whose entries are all 0. F(A) is also symmetric. Prove that if A  if A ∈ O(n), then DF(A) : Mn×n → S n×n is onto. (S is the set of symmetric

n×n matrix, M is the set of n×n matrix)