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\begin_body

\begin_layout Title
Rešitev osme domače naloge Linearne Algebre
\end_layout

\begin_layout Author

\noun on
Anton Luka Šijanec
\end_layout

\begin_layout Date
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
today
\end_layout

\end_inset


\end_layout

\begin_layout Abstract
Za boljšo preglednost sem svoje rešitve domače naloge prepisal na računalnik.
 Dokumentu sledi še rokopis.
 Naloge je izdelala asistentka Ajda Lemut.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
newcommand
\backslash
euler{e}
\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
Dokaži, da je 
\begin_inset Formula $\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw$
\end_inset

 skalarni produkt in ugotovi, ali je
\begin_inset Formula 
\[
A=\left[\begin{array}{ccc}
0 & 2 & -2\\
0 & 1 & 0\\
1 & 2 & -1
\end{array}\right]
\]

\end_inset

 normalna preslikava glede na 
\begin_inset Formula $\left[\cdot,\cdot\right]$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Paragraph
Rešitev
\end_layout

\begin_layout Standard
Predpostavljam polje 
\begin_inset Formula $\mathbb{R}$
\end_inset

 in vektorski prostor 
\begin_inset Formula $V=\mathbb{R}^{3}$
\end_inset

.
 
\begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$
\end_inset

 je skalarni produkt, če zadošča naslednjim lastnostim.
 Dokažimo jih za 
\begin_inset Formula $\left[\cdot,\cdot\right]$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\forall v\in V:v\not=0\Rightarrow\langle v,v\rangle>0$
\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $\forall v,u\in V:\langle v,u\rangle=\langle u,v\rangle$
\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $\forall\alpha_{1},\alpha_{2}\in\mathbb{C}\forall u_{1},u_{2},v\in V:\langle\alpha_{1}v_{1}+\alpha_{2}v_{2},v\rangle=\alpha_{1}\langle u_{1},v\rangle+\alpha_{2}\langle u_{2},v\rangle$
\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
Pokaži 
\begin_inset Formula $A:V\to V$
\end_inset

 je normalna 
\begin_inset Formula $\Leftrightarrow AA^{*}-A^{*}A$
\end_inset

 je pozitivno semidefinitna.
\end_layout

\begin_layout Enumerate
Naj bo 
\begin_inset Formula $w_{1}=\left(1,1,1,1\right)$
\end_inset

, 
\begin_inset Formula $w_{2}=\left(3,3,-1,-1\right)$
\end_inset

 in 
\begin_inset Formula $w_{3}=\left(6,0,2,0\right)$
\end_inset

.
\end_layout

\begin_layout Enumerate
Poišči singularni razcep matrike
\begin_inset Formula 
\[
A=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -2 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]\text{.}
\]

\end_inset


\end_layout

\begin_deeper
\begin_layout Paragraph
Rešitev
\end_layout

\begin_layout Itemize
Iščemo 
\begin_inset Formula $U$
\end_inset

, 
\begin_inset Formula $\Sigma$
\end_inset

 in 
\begin_inset Formula $V$
\end_inset

, da velja 
\begin_inset Formula $A=U\Sigma V^{*}$
\end_inset

.
\end_layout

\begin_layout Itemize
Diagonalci 
\begin_inset Formula $\Sigma$
\end_inset

 so singularne vrednosti 
\begin_inset Formula $A$
\end_inset

.
 Singularne vrednosti 
\begin_inset Formula $A$
\end_inset

 so koreni lastnih vrednosti
\end_layout

\begin_layout Itemize
\begin_inset Formula $A^{*}A$
\end_inset

, torej 
\begin_inset Formula $\sigma_{1}=2$
\end_inset

, 
\begin_inset Formula $\sigma_{2}=1$
\end_inset

, 
\begin_inset Formula $\sigma_{3}=0$
\end_inset

.
\begin_inset Formula 
\[
A^{*}A=\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & -2 & 0 & 0\\
0 & 0 & 0 & 0
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -2 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 0
\end{array}\right]
\]

\end_inset


\begin_inset Formula 
\[
\Sigma=\left[\begin{array}{ccc}
\sigma_{1} & 0 & 0\\
0 & \sigma_{2} & 0\\
0 & 0 & \sigma_{3}\\
0 & 0 & 0
\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Itemize
Stolpci 
\begin_inset Formula $V$
\end_inset

 so ortonormirana baza jedra 
\begin_inset Formula $A^{*}A-\sigma^{2}I$
\end_inset

 za vse singularne vrednosti 
\begin_inset Formula $\sigma$
\end_inset

.
\begin_inset Formula 
\[
A^{*}A-4I=\left[\begin{array}{ccc}
-3 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]\Rightarrow x=0\Rightarrow v_{1}=\left(0,1,0\right)
\]

\end_inset


\begin_inset Formula 
\[
A^{*}A-1I=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 3 & 0\\
0 & 0 & 0
\end{array}\right]\Rightarrow y=0\Rightarrow v_{2}=\left(1,0,0\right)
\]

\end_inset


\begin_inset Formula 
\[
A^{*}A-0I=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 0
\end{array}\right]\Rightarrow x=y=0\Rightarrow v_{3}=\left(0,0,1\right)
\]

\end_inset


\begin_inset Formula 
\[
V=\left[\begin{array}{ccc}
v_{1} & v_{2} & v_{3}\end{array}\right]=\left[\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 1
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Itemize
Stolpci 
\begin_inset Formula $U$
\end_inset

 so ortonormirana baza in velja 
\begin_inset Formula $\forall i\in\left\{ 1..\rang A\right\} :u_{i}=\sigma_{i}^{-1}Av_{i}$
\end_inset

.
 Stolpične vektorje 
\begin_inset Formula $v_{\rang A+1},\dots,v_{m}$
\end_inset

 najdemo tako, da dopolnimo 
\begin_inset Formula $v_{1},\dots,v_{\rang A}$
\end_inset

 do ONB.
\begin_inset Formula 
\[
U=\left[\begin{array}{cccc}
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Itemize
Dobljene matrike zmnožimo, s čimer potrdimo veljavnost singularnega razcepa:
\begin_inset Formula 
\[
U\Sigma V^{*}=\left[\begin{array}{cccc}
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]\left[\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -2 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]=A
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja.
 Zavedam se namreč njihovega neličnega izgleda.
\end_layout

\end_body
\end_document