Prove that $\|Q\mathbf{v}\|=\|\mathbf{v}\|$

Question

Prove that if $Q$ is a real $n\times n$ orthogonal matrix and $\mathbf{v}$ is in $\mathbb{R}^{n}$, then $$\|Q\mathbf{v}\| = \|\mathbf{v}\|.$$ Be sure to set out your arguments clearly and logically, giving full reasons.

Hello all, To solve this do I try and show that $1$ is an eigenvalue? Or is there some rule I'm missing.

Answer 1

That wouldn't be enough, since this needs to be true for any $\mathbf{v}$, not just for the eigenvector. Instead, consider $||Q\mathbf{v}||^2 = \mathbf{v}^T Q^T Q \mathbf{v}$, and see what you'd get.

Answer 2

Notice that for all $x$ and $y$ column vectors of $\mathbb{R}^n$, one has: $$\langle x,y\rangle={}^\intercal xy.$$ Therefore, for all $v\in\mathbb{R}^n$, one has: $$\|Qv\|^2=\langle Qv,Qv\rangle={}^\intercal v{}^\intercal QQv={}^\intercal vv=\|v\|^2,$$ since $Q$ is orthogonal i.e. ${}^\intercal QQ=I_n$. Whence the result.