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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-03 18:51:36 +0200 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-03 18:51:36 +0200 |
commit | 680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d (patch) | |
tree | aff901d861b84c62e8eb2b7ff9981ae13a3f5baa /vorlesungen/slides/2/hilbertraum/spektral.tex | |
parent | add slides (diff) | |
download | SeminarMatrizen-680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d.tar.gz SeminarMatrizen-680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d.zip |
phases
Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/spektral.tex')
-rw-r--r-- | vorlesungen/slides/2/hilbertraum/spektral.tex | 62 |
1 files changed, 33 insertions, 29 deletions
diff --git a/vorlesungen/slides/2/hilbertraum/spektral.tex b/vorlesungen/slides/2/hilbertraum/spektral.tex index b7a44f8..b561b69 100644 --- a/vorlesungen/slides/2/hilbertraum/spektral.tex +++ b/vorlesungen/slides/2/hilbertraum/spektral.tex @@ -19,69 +19,73 @@ Hilbertraum $H$ $A\colon H\to H$ linear \end{itemize} \end{block} +\uncover<2->{% \begin{block}{Eigenwerte} $x\in H$ ein EV von $A$ zum EW $\lambda\ne 0$ \begin{align*} -\langle x,x\rangle +\uncover<3->{\langle x,x\rangle &= \frac1{\lambda} -\langle x,\lambda x\rangle -= +\langle x,\lambda x\rangle} +\uncover<3->{= \frac1{\lambda} -\langle x,Ax\rangle +\langle x,Ax\rangle} \\ -&= +&\uncover<4->{= \frac1{\lambda} -\langle Ax,x\rangle -= +\langle Ax,x\rangle} +\uncover<5->{= \frac{\overline{\lambda}}{\lambda} -\langle x,x\rangle +\langle x,x\rangle} \\ -\frac{\overline{\lambda}}{\lambda}&=1 -\quad\Rightarrow\quad -\overline{\lambda} = \lambda +\uncover<6->{\frac{\overline{\lambda}}{\lambda}&=1 \quad\Rightarrow\quad -\lambda\in\mathbb{R} +\overline{\lambda} = \lambda} +\uncover<7->{\quad\Rightarrow\quad +\lambda\in\mathbb{R}} \end{align*} -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<8->{% \begin{block}{Orthogonalität} $u,v$ EV zu EW $\mu,\lambda\in \mathbb{R}\setminus\{0\}$, $\overline{\mu}=\mu\ne\lambda$ \begin{align*} +\uncover<9->{ \langle u,v\rangle &= \frac{1}{\mu} -\langle \mu u,v\rangle -= +\langle \mu u,v\rangle} +\uncover<10->{= \frac{1}{\mu} -\langle Au,v\rangle +\langle Au,v\rangle} \\ -&= +&\uncover<11->{= \frac{1}{\mu} -\langle u,Av\rangle -= +\langle u,Av\rangle} +\uncover<12->{= \frac{1}{\mu} -\langle u,\lambda v\rangle -= +\langle u,\lambda v\rangle} +\uncover<13->{= \frac{\lambda}{\mu} -\langle u,v\rangle +\langle u,v\rangle} \\ -\Rightarrow +\uncover<14->{\Rightarrow \; 0 &= \underbrace{\biggl(\frac{\lambda}{\mu}-1\biggr)}_{\displaystyle \ne 0} -\langle u,v\rangle -\;\Rightarrow\; -\langle u,v\rangle = 0 +\langle u,v\rangle} +\uncover<15->{\;\Rightarrow\; +\langle u,v\rangle = 0} \end{align*} -EV zu verschiedenen EW sind orthogonal -\end{block} +\uncover<16->{EV zu verschiedenen EW sind orthogonal} +\end{block}} \end{column} \end{columns} +\uncover<17->{% \begin{block}{Spektralsatz} Es gibt eine Hilbertbasis von $H$ aus Eigenvektoren von $A$ -\end{block} +\end{block}} \end{frame} \egroup |