Let $T$ be a linear operator on a finite dimensional inner product space $V$ such that it is both self-adjoint and unitary.Then prove that it can only arise from the subspace $W$ s.t.
$$V=W \bigoplus W^{\perp}$$
and if $a=a_1+a_2$ then $T(a)=a_1-a_2$.
Now if $T$ is self-adjoint then $T=T^*$ for an orthodontist basis of $V$.Also, $TT^*=T^*T=I$ and $T$ carries orthonormal basis vectors to orthonormal basis vectors if $T^*$ is unitary.
now from here we can conclude that $$T^2=I$$ .Hence $T$ is an idempotent linear operator. since $T$ is an idempotent linear operator then we see that,
$$V = R(T) \bigoplus N(T)$$ Now we call R(T) to be as $W$.
We need to show that $N(T)=R(T)^{\perp}$ .
let $x \in N(T)$.Then, we take a vector $y \in R(T)$ ,we see that
$$\langle y, T^*T(x)\rangle=\langle y, T^*(0)\rangle=\langle y, 0\rangle=0\,.$$
Then, it is easy to conclude that $N(T)=W^{\perp}$.
Now how do I proceed from here.
This from Hoffman exercise -8.4 no-12
I would prefer some hints and intuitions behind those hints instead of complete answers.
