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author | LordMcFungus <mceagle117@gmail.com> | 2021-03-22 18:05:11 +0100 |
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committer | GitHub <noreply@github.com> | 2021-03-22 18:05:11 +0100 |
commit | 76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7 (patch) | |
tree | 11b2d41955ee4bfa0ae5873307c143f6b4d55d26 /vorlesungen/slides/2/frobeniusnorm.tex | |
parent | more chapter structure (diff) | |
parent | add title image (diff) | |
download | SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.tar.gz SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.zip |
Merge pull request #1 from AndreasFMueller/master
update
Diffstat (limited to 'vorlesungen/slides/2/frobeniusnorm.tex')
-rw-r--r-- | vorlesungen/slides/2/frobeniusnorm.tex | 96 |
1 files changed, 96 insertions, 0 deletions
diff --git a/vorlesungen/slides/2/frobeniusnorm.tex b/vorlesungen/slides/2/frobeniusnorm.tex new file mode 100644 index 0000000..461005a --- /dev/null +++ b/vorlesungen/slides/2/frobeniusnorm.tex @@ -0,0 +1,96 @@ +% +% 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} |