Q:

orthogonal projection and inner product

Let $P\in \mathbb{R}^{d\times d}$ be an orthogonal projection matrix, and $x,y\in\mathbb{R}^d$ are two arbitrary vectors. I was wondering if the following inequality hold:
$2\langle x,z\rangle\cdot \langle Px,Pz\rangle\ge \langle Px,Pz\rangle^2$.
At first, I thought I can prove it immediately from some inequality (e.g., Cauchy-Schwarz). But it seems more difficult than I expected. Any help will be appreciated.

A:

False. In $\Bbb R^2$, consider $\renewcommand\vec[1]{{\boldsymbol #1}}\vec{x} = (1,1)$, $\vec{z} = (-1,1)$ and $P$ the orthogonal projection in the axis $y = 0$. Then $$2\langle \vec{x},\vec{z}\rangle \langle P\vec{x},P\vec{z}\rangle = 0 \color{red}{<} \langle P\vec{x},P\vec{z}\rangle^2 = \langle (1,0),(-1,0) \rangle^2 =1.$$

