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.

Prove that A ∈ O(n) if and only if (Av) · (Aw) = v · w for all v, w ∈ R n . Remark: This set O(n) is called the orthogonal group in n dimensions, and its elements are called orthogonal matrices. You have probably seen them in linear algebra. This exercise indicates a reason why O(n) is a very important object in many branches of both mathematics and mathematical physics.