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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/5/spektrum.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/5/spektrum.tex')
-rw-r--r-- | vorlesungen/slides/5/spektrum.tex | 76 |
1 files changed, 76 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/spektrum.tex b/vorlesungen/slides/5/spektrum.tex new file mode 100644 index 0000000..6cbdd7f --- /dev/null +++ b/vorlesungen/slides/5/spektrum.tex @@ -0,0 +1,76 @@ +% +% spektrum.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spektrum} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +$A\colon V\to V$ beschränkter Operator zwischen Banach-Räumen +\[ +\operatorname{Sp}A += +\left\{ +\lambda\in\mathbb{C} +\;\left|\; +\begin{minipage}{2cm}\raggedright +$A-\lambda I$ nicht invertierbar +\end{minipage} +\right. +\right\} +\] +\end{block} +\uncover<2->{% +\begin{block}{Endlichdimensionale Räume} +\vspace{-15pt} +\begin{align*} +&\lambda\in\operatorname{Sp}A +\\ +\uncover<3->{ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar} +} +\\ +\uncover<4->{ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär} +} +\\ +\uncover<5->{ +\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0 +} +\\ +\uncover<6->{ +\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v +} +\end{align*} +\uncover<7->{% +$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte +} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Unendlichdimensional} +Es gibt eine Folge $x_n\in V$ von Einheitsvektoren +$\|x_n\|=1$ +mit +\begin{align*} +\lim_{n\to\infty} (A - \lambda)x_n &= 0 +\end{align*} +\end{block}} +\uncover<9->{% +\begin{block}{Spektrum und Norm} +\[ +\operatorname{Sp}(A) +\subset +\{\lambda\in\mathbb{C}\;|\; +|\lambda|\le \|A\|\} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} |