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Let $‖\cdot ‖$ be a norm invariant under unitary trasformations.
Is it true that
$‖\left(\begin{array}{c}\stackrel{^}{L}-L\\ G\end{array}\right)‖$
is minimized when $\stackrel{^}{L}=L$ ($\stackrel{^}{L}$ and $G$ are fixed, $L$ is the only variable quantity), i.e $‖\left(\begin{array}{c}\stackrel{^}{L}-L\\ G\end{array}\right)‖=‖\left(\begin{array}{c}0\\ G\end{array}\right)‖$ ?
I'm unable to prove using just elementar inequalities such as triangle inequality or norm properties, any help would be appreciated.
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Kroatujon3
If the transformation $U$ that maps $\left[a,b{\right]}^{T}$ into $\left[-a,b{\right]}^{T}$ is unitary (it might depend on your exact setting), then :
$\begin{array}{rl}‖\frac{x+Ux}{2}‖& \le \frac{1}{2}\left(‖x‖+‖Ux‖\right)\\ & =‖x‖\end{array}$
but if $x=\left[a,b{\right]}^{T}$ then $\frac{x+Ux}{2}=\left[0,b{\right]}^{T}$.
Therefore $\underset{a}{min}‖\left[a,b{\right]}^{T}‖=‖\left[0,b{\right]}^{T}‖$. This can then be adapted by translation to your setting.