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%
% frobeniusnorm.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Frobenius-Norm}
\vspace{-15pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Skalarprodukt}
$A,B\in M_{m\times n}(\mathbb{C})$
\begin{align*}
\langle A,B\rangle_F
&\uncover<2->{=
\sum_{i,j} \overline{a}_{ik}b_{ik}}
\uncover<3->{=
\operatorname{Spur} A^*B}
\\
\uncover<4->{
\|A\|_F^2
&=
\langle A,A\rangle}
\uncover<5->{=
\sum_{i,k} |a_{ik}|^2}
\end{align*}
\uncover<6->{%
$\Rightarrow M_{m\times n}(\mathbb{C})$ ist ein normierter Raum}
\end{block}
\end{column}
\begin{column}{0.48\textwidth}
\uncover<12->{%
\begin{block}{Singulärwertzerlegung}
\vspace{-12pt}
\begin{align*}
\uncover<13->{
A
&=
U\Sigma V^*}
\\
\uncover<14->{
A^*A
&=
V\Sigma^*U^*U\Sigma V^*}
\uncover<15->{=
V\Sigma^*\Sigma V^*}
\\
\uncover<16->{%
\operatorname{Spur}{A^*A}
&=
\operatorname{Spur}V\Sigma^*\Sigma V^*}
\\
\uncover<17->{%
&=
\operatorname{Spur}V^*V\Sigma^*\Sigma}
\\
\uncover<18->{%
&=
\operatorname{Spur}\Sigma^*\Sigma}
\uncover<19->{=
\sum_{i} |\sigma_i|^2}
\end{align*}
\end{block}}
\end{column}
\end{columns}
\uncover<7->{%
\begin{block}{Produkt}
\vspace{-10pt}
\begin{align*}
\|AB\|_F
\uncover<8->{=
\sum_{i,j}
\biggl|
\sum_{k}
a_{ik}b_{kj}
\biggr|^2}
&\uncover<9->{\le
\sum_{i,j}
\biggl(
\sum_k |a_{ik}|^2
\biggr)
\biggl(
\sum_l |b_{lj}|^2
\biggr)}
\\
\uncover<10->{
&=
\sum_{i,k} |a_{ik}|^2
\sum_{l,j} |b_{lj}|^2}
\uncover<11->{=
\|A\|_F\cdot \|B\|_F}
\end{align*}
\end{block}}
\end{frame}
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