The problem:
Show that if $\textbf{0} \neq \textbf{v} \in \mathbb{R}^{n}$ and $E \in \mathbb{R}^{n\times n}$, then
$$\Big\lVert E(I - \frac{\textbf{v}\textbf{v}^{T}}{\textbf{v}^{T}\textbf{v}})\Big\rVert_{F}^{2} = \|E\|_{F}^{2} - \frac{\|E \textbf{v}\|_{2}^{2}}{\textbf{v}^{T}\textbf{v}}$$
What I have tried:
LHS = $\Big\|E - E\frac{\textbf{v}\textbf{v}^{T}}{\textbf{v}^{T}\textbf{v}}\Big\|_{F}^{2}$
and I want to expand it using $ \|A\|_{F}^{2} = \mathrm{tr}(AA^{T})$.
But it seems too complicated, am I on the right track? Or could there be any more efficient ways?
Thank you very much!
It's not that hard, maybe safe yourself some notation first. Set $P:=vv^T/v^Tv$, then $$ \|E-EP\|_F^2 = \mathrm{tr}[(E-EP)(E-EP)^T] = \mathrm{tr}(EE^T-EP^TE^T-EPE^T+EPP^TE^T). $$ Now as already pointed out, $P$ is an orthogonal projection, that is, $P^T=P$ and $P^2=PP^T=P$. Using it above gives $$ \begin{split} \|E-EP\|_F^2 &= \mathrm{tr}(EE^T-EPE^T-EPE^T+EPE^T)\\ &= \mathrm{tr}(EE^T-EPE^T) = \mathrm{tr}[EE^T-EP(EP)^T]\\ &= \mathrm{tr}(EE^T)-\mathrm{tr}[EP(EP)^T] =\|E\|_F^2 - \|EP\|_F^2 \end{split} $$ Finally, $$ \begin{split} \|EP\|_F^2&=\mathrm{tr}[(EP)^TEP]=\mathrm{tr}(P^TE^TEP)\\ &=\mathrm{tr}(vv^TE^TEvv^T)/(v^Tv)^2=\frac{v^TE^TEv\cdot v^Tv}{(v^Tv)^2} = \frac{\|Ev\|_2^2}{v^Tv}. \end{split} $$