Let $A$ be a real $n \times n$ matrix. I'm trying to prove that if $A$ maps orthogonal vectors into orthogonal vectors and $\lVert A \rVert = 1$, then, for every $x,y \in \mathcal{l}_2^n$, $(Ax,Ay)=(x,y)$.
Obviously $\mathcal{l}_2^n$ is $\mathbb{R}^n$ with the Euclidean norm, and $(x,y)$ denotes the inner product of $x$ and $y$.
I have proven that if $\{e_k\}_{k=1}^n$ is the canonical basis, then $Ae_t$ is orthogonal to $Ae_r$ whenever $t \neq r$. All I need to finish my proof is that $(Ae_k,Ae_k)=1$, since that would imply that the product of the transpose of $A$ and $A$ is the identity matrix. I've tried to use the fact that $\lVert A \rVert = 1$ to prove this, but I haven't made any progress.
Any ideas?