Behaviour of orthogonal matrices

Question

I am given that A is an orthogonal matrix of order $n$, and $u, v$ are Vectors in the $R^n $ space.

I need to prove that $||u|| = ||Au||$. The first step of the solution hint I am given is that $$||Au||^2 = (Au)^T(Au)$$. Why is this so? I know that $A^{-1} = A^T$ in the definition of an orthogonal matrix, but how does this contribute to the above statement? Or is there some other property I'm missing out on?

Answer 1

Assume that $A$ is orthogonal, i.e. that $A^T A = I$ (observe that this is equivalent to $A^{-1} = A^T$). We consider

$$ || A x ||^2 = (Ax)^T (Ax) = x^T A^T A x = x^T x = || x || ^2, $$

which shows that $|| A x || = || x ||$.

Comment: On advise from another user, I posted a more thorough version of my earlier comment as this answer.

Answer 2

If $u=(u_1,u_2,\ldots,u_n)$, then\begin{align}u^\top u&=\begin{pmatrix}u_1\\u_2\\\vdots\\u_n\end{pmatrix}(u_1,u_2,\ldots,u_n)\\&=u_1^{\,2}+u_2^{\,2}+\cdots+u_n^{\,2}\\&=\|u\|^2.\end{align}