TITLE: If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$
QUESTION [8 upvotes]: Is this statment true: For any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$, if $A$ is orthogonal.
I was working on a proof for my linear algebra class, when I noticed that the entire proof could be reduced to simple algebraic work conditional on the following statement being true:

$$\text{For any vector $x$ such that $Ax = b$, $\Vert x \Vert_2 = \Vert b \Vert_2$, if $A$ is orthogonal.}$$

I have not found a straight answer to this question. I believe the statement is true but I am not sure about how to prove it.
Thanks in advance.

REPLY [5 votes]: Recall that $A$ is orthogonal if $A^TA = I$. Now look at $\Vert b \Vert_2^2$. Hence, we get that
$$\Vert b \Vert_2^2 = b^Tb = (Ax)^T (Ax) = x^T\underbrace{A^TA}_Ix = x^Tx = \Vert x \Vert_2^2$$