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.



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 $$||Px ||^2 = (Px)^T(Px) = x^T P^T P x = x^Tx = ||x||^2.$$

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