I rewrote a vector equation and am searching for feedback if I did it properly.
I start with:
$\sum_{k=1}^N \lvert{<v,e_k>}\rvert^2=\alpha \Vert{v}\Vert^2$
where $v,e_k \in \mathbb{R}^5$ and $\alpha \in \mathbb{R}$. I want to have a simple way to show that the equality above holds for a given set of $e_k$, independent of $v$.
I do the following:
$<=>\sum_{k=1}^N \lvert{<v,e_k>}\rvert^2=\alpha \Vert{v}\Vert^2$
$<=>\sum_{k=1}^N (v*e_k^T)^2=\alpha \sqrt{v^2}^2$
$<=>v^2 \sum_{k=1}^N (e_k^T) ^2 =\alpha v^2$
$<=>\sum_{k=1}^N (e_k^T) ^2 = \alpha$
Is this correct? I am not 100% certain about the way I move $v$ around given it is a vector that does not commute necessarily.
In a compact form: $\alpha = \sum_{k=1}^N \Vert{e_k}\Vert^2 \cos^2(\phi_k)$ where $\phi_k$ is the angle between the vectors $v$ and $e_k$.