If $P$ is an $n\times n$ orthogonal matrix, then prove that $\Vert Px\Vert = \Vert x\Vert$
I tried manipulating the expression arbitrarily, but I can never understand why should I be doing whatever I did. Why does it work? And most importantly, how do you know what initiatives to take?
I raised both sides to 2, and I got:
$$P\bar{x}\cdot P\bar{x} = \bar{x}\cdot\bar{x}$$ This is where I get caught off; I transposed part of the expression to get:
$$\bar{x}^{T}P^{T}P\bar{x} = \bar{x}\cdot\bar{x}$$
Using associativity, P transpose and P yields the identity. But then what? All I get is
$$\bar{x}^{T}\bar{x} = \bar{x}\cdot\bar{x}$$
I don't even know if I'm heading off properly.
Thanks