Prove $||Px|| = ||x||$ for Orthogonal Matrices

Question

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

Answer 1

You identified the properties you need to manipulate, but you should never start with the conclusion (i.e. the equality) and reduce it down to a trivial statement. It logically makes no sense, unless every implication you made was bidirectional. Instead, write $$||Px ||^2 = (Px)^T(Px) = x^T P^T P x = x^Tx = ||x||^2.$$