 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/p/p4/a/solution

Let

P = v v^{T} / (v^{T} v)

, which is a nonzero symmetric orthogonal projection:

P^{T} = P

and

P^{2} = P

. Then

H_{v, α} = I - α P

is symmetric and

H_{v, α}^{T} H_{v, α} = (I - α P)^{2} = I + (α^{2} - 2 α) P .

Since

P \neq = 0

, this equals the identity exactly when

α (α - 2) = 0

. Thus

H_{v, α} is orthogonal ⟺ α \in {0, 2} .

 New to topics? Read the docs here!