If $Q$ is hermitian and unitary, prove that $Q=I-2P$ for some orthogonal projection $P$?

Question

Good afternoon everyone.

I am in a numerical linear algebra class, and have come across this problem:

Prove that for $V$ a subspace of $\mathbb{C}^m$, $P$ the orthogonal projector onto $V$, and $Q=I-2P$,
$Q=Q^*$ ($Q$ is hermitian) and $Q^*Q=QQ^*=I$ ($Q$ is unitary)

and conversely,

Prove that
$\forall Q \in \mathbb{C}^{m\times m}. ((Q=Q^*)\land (Q^*Q=I) \implies \exists P \in \mathbb{C}^{m\times m}. (P=P^*) \land (Q=I-2P))$

or in other words

If $Q$ is both hermitian and unitary then $Q=I-2P$ for some orthogonal projector $P$.

I have succeeded in proving the first part of the problem by using some properties of the conjugate transpose (namely $(A+B)^*=A^*+B^*$) and some simple algebra, but the second one puzzles me. My paper literally just looks like this right now:

7. $Q=Q^*$, $Q^*Q=I$
   $Q^2=I$

And I'm stuck. Do I proceed by supposing a projection onto $V$? Do I somehow prove that $P$ is an orthogonal projector?

Thanks much.

Answer 1

$Q^\dagger = Q; \tag 1$

$Q^\dagger = Q^{-1}; \tag 2$

$Q = Q^\dagger = Q^{-1} \Longrightarrow Q^2 = I; \tag 3$

set

$P = \dfrac{1}{2}(I - Q); \tag 4$

then

$P^2 = \dfrac{1}{4}(I - Q)^2$ $= \dfrac{1}{4}(I - 2Q + Q^2) = \dfrac{1}{4}(2 - 2Q) = \dfrac{1}{2}(I - Q) = P, \tag 5$

showing that $P$ is a projection; also,

$P^\dagger = \dfrac{1}{2}(I - Q)^\dagger = \dfrac{1}{2}(I^\dagger - Q^\dagger) = \dfrac{1}{2}(I - Q) = P; \tag 6$

$P$ is thus hermitian, hence for

$Py = 0, \; z = Pw, \tag 7$

$\langle y, z \rangle = \langle y, Pw \rangle = \langle P^\dagger y, w \rangle = \langle Py, w \rangle = \langle 0, w \rangle = 0, \tag 8$

showing that $\ker P$ and $\text{range}(P)$ are orthogonal, which is the definition of orthogonal projection.

Finally, we note that (4) implies

$Q = I - 2P. \tag{9}$

Answer 2

Put $P=\frac{1}{2}(I-Q)$ (i.e., isolate $P$ in the given expression), and let $V$ be the range of $P$. Then it is readily verified that $P^*=P^2=P$, so $P$ is an orthogonal projection onto its range.