#LyX 2.3 created this file. For more info see http://www.lyx.org/
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enumitem
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\language slovene
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\end_header

\begin_body

\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 Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{multicols}{2}
\end_layout

\end_inset


\end_layout

\begin_layout Paragraph
Drobnarije od prej
\end_layout

\begin_layout Standard
\begin_inset Formula $\det A=\det A^{T}$
\end_inset


\end_layout

\begin_layout Standard
Vsota je direktna 
\begin_inset Formula $\Leftrightarrow V\cap U=\left\{ 0\right\} $
\end_inset


\end_layout

\begin_layout Paragraph
Skalarni produkt
\end_layout

\begin_layout Standard
\begin_inset Formula $\left\langle v,v\right\rangle >0$
\end_inset

, 
\begin_inset Formula $\left\langle v,u\right\rangle =\overline{\left\langle u,v\right\rangle }$
\end_inset

, 
\begin_inset Formula $\left\langle \alpha_{2}u_{1}+\alpha_{2}u_{2},v\right\rangle =\alpha_{1}\left\langle u_{1},v\right\rangle +\alpha_{2}\left\langle u_{2},v\right\rangle $
\end_inset

, 
\begin_inset Formula $\left\langle u,\alpha_{1}v_{1}+\alpha_{2}v_{2}\right\rangle =\overline{\alpha_{1}}\left\langle u,v_{1}\right\rangle +\overline{\alpha_{2}}\left\langle u,v_{2}\right\rangle $
\end_inset


\end_layout

\begin_layout Standard
Standardni: 
\begin_inset Formula $\left\langle \left(\alpha_{1},\dots,\alpha_{n}\right),\left(\beta_{1},\dots,\beta_{n}\right)\right\rangle =\alpha_{1}\overline{\beta_{1}}+\cdots\alpha_{n}\overline{\beta_{n}}$
\end_inset


\end_layout

\begin_layout Standard
Norma: 
\begin_inset Formula $\left|\left|v\right|\right|^{2}=\left\langle v,v\right\rangle $
\end_inset

: 
\begin_inset Formula $\left|\left|v\right|\right|>0\Leftrightarrow v\not=0$
\end_inset

, 
\begin_inset Formula $\left|\left|\alpha v\right|\right|=\left|\alpha\right|\left|\left|v\right|\right|$
\end_inset


\end_layout

\begin_layout Standard
Trikotniška neenakost: 
\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$
\end_inset


\end_layout

\begin_layout Standard
Cauchy-Schwarz: 
\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|\leq\left|\left|v\right|\right|\left|\left|u\right|\right|$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $v\perp u\Leftrightarrow\left\langle u,v\right\rangle =0$
\end_inset

.
 
\begin_inset Formula $M$
\end_inset

 ortog.
 
\begin_inset Formula $\Leftrightarrow\forall u,v\in M:v\perp u\wedge v\not=0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $M$
\end_inset

 normirana 
\begin_inset Formula $\Leftrightarrow\forall u\in M:\left|\left|u\right|\right|=1$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $M$
\end_inset

 ortog.
 
\begin_inset Formula $\Rightarrow M$
\end_inset

 lin.
 neod., Ortog.
 baza 
\begin_inset Formula $\sim$
\end_inset

 ortog.
 ogrodje
\end_layout

\begin_layout Standard
\begin_inset Formula $v\perp M\Leftrightarrow\forall u\in M:v\perp u$
\end_inset


\end_layout

\begin_layout Paragraph
Fourierov razvoj
\end_layout

\begin_layout Standard
\begin_inset Formula $v_{i}$
\end_inset

 ortog.
 baza za 
\begin_inset Formula $V$
\end_inset

, 
\begin_inset Formula $v\in V$
\end_inset

 poljuben.
 
\begin_inset Formula $v=\sum_{i=1}^{n}\frac{\left\langle v,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
\end_inset


\end_layout

\begin_layout Standard
Parsevalova identiteta: 
\begin_inset Formula $\left|\left|v\right|\right|^{2}=\sum_{i=1}^{n}\frac{\left|\left\langle v,v_{i}\right\rangle \right|^{2}}{\left\langle v_{i},v_{i}\right\rangle }$
\end_inset


\end_layout

\begin_layout Paragraph
Projekcija na podprostor
\end_layout

\begin_layout Standard
let 
\begin_inset Formula $V$
\end_inset

 podprostor 
\begin_inset Formula $W$
\end_inset

.
 
\begin_inset Formula $v'$
\end_inset

 je ortog.
 proj vektorja 
\begin_inset Formula $v$
\end_inset

 
\begin_inset Formula $\Leftrightarrow\forall w\in W:\left|\left|v-v'\right|\right|\leq\left|\left|v-w\right|\right|\sim\text{v'}$
\end_inset

 je najbližje 
\begin_inset Formula $V$
\end_inset

 izmed elementov 
\begin_inset Formula $W$
\end_inset

.
 
\begin_inset Formula $\sun$
\end_inset

 Pitagora:
\end_layout

\begin_layout Standard
Zadošča preveriti ortogonalnost 
\begin_inset Formula $v-v'$
\end_inset

 na vse elemente 
\begin_inset Formula $W$
\end_inset

.
\end_layout

\begin_layout Standard
Formula za ort.
 proj.: 
\begin_inset Formula $v'=\sum_{i=0}^{n}\frac{\left\langle v,w_{i}\right\rangle }{\left\langle w_{i},w_{i}\right\rangle }$
\end_inset

, kjer je 
\begin_inset Formula $w_{i}$
\end_inset

 OB 
\begin_inset Formula $W$
\end_inset

.
\end_layout

\begin_layout Paragraph
Obstoj ortogonalne baze (Gram-Schmidt)
\end_layout

\begin_layout Standard
let 
\begin_inset Formula $\left\{ u_{1},\dots,u_{n}\right\} $
\end_inset

 baza 
\begin_inset Formula $V$
\end_inset

.
 Zanj konstruiramo OB 
\begin_inset Formula $\left\{ v_{1},\dots,v_{n}\right\} $
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $v_{1}=u_{1}$
\end_inset

, 
\begin_inset Formula $v_{2}=u_{2}-\frac{\left\langle u_{2},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
\end_inset

, 
\begin_inset Formula $v_{3}=u_{3}-\frac{\left\langle u_{3},v_{2}\right\rangle }{\left\langle v_{2},v_{2}\right\rangle }v_{2}-\frac{\left\langle u_{3},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
\end_inset

...
 
\begin_inset Formula $v_{k}=u_{k}-\sum_{i=1}^{k-1}\frac{\text{\left\langle u_{k},v_{i}\right\rangle }}{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
\end_inset


\end_layout

\begin_layout Paragraph
Ortogonalni komplement
\end_layout

\begin_layout Standard
let 
\begin_inset Formula $S\subseteq V$
\end_inset

.
 
\begin_inset Formula $S^{\perp}=\left\{ v\in V;v\perp S\right\} $
\end_inset

.
 Velja: 
\begin_inset Formula $S^{\perp}$
\end_inset

 podprostor 
\begin_inset Formula $V$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $W$
\end_inset

 podprostor 
\begin_inset Formula $V$
\end_inset

.
 Velja: 
\begin_inset Formula $W\oplus W^{\perp}=V$
\end_inset

 in 
\begin_inset Formula $\left(W^{\perp}\right)^{\perp}=W$
\end_inset

.
\end_layout

\begin_layout Standard
Če je 
\begin_inset Formula $\left\{ u_{1},\dots,u_{k}\right\} $
\end_inset

 OB podprostora 
\begin_inset Formula $V$
\end_inset

, je dopolnitev do baze vsega 
\begin_inset Formula $V^{\perp}$
\end_inset

.
\end_layout

\begin_layout Standard
Za vektorske podprostore 
\begin_inset Formula $V_{i}$
\end_inset

 VPSSP 
\begin_inset Formula $W$
\end_inset

 velja:
\end_layout

\begin_layout Standard
\begin_inset Formula $S\subseteq W\Rightarrow\left(S^{\perp}\right)^{\perp}=\mathcal{L}in\left\{ S\right\} $
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $V_{1}\subseteq V_{2}\Rightarrow V_{2}^{\perp}\subseteq V_{1}^{\perp}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\left(V_{1}+v_{2}\right)^{\perp}=V_{1}^{\perp}\cup V_{2}^{\perp}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\left(V_{1}\cap V_{2}\right)^{\perp}=V_{1}^{\perp}+V_{2}^{\perp}$
\end_inset


\end_layout

\begin_layout Paragraph
Linearni funkcional
\end_layout

\begin_layout Standard
je linearna preslikava 
\begin_inset Formula $V\to F$
\end_inset

, če je 
\begin_inset Formula $V$
\end_inset

 nad poljem 
\begin_inset Formula $F$
\end_inset

.
\end_layout

\begin_layout Standard
Rieszov izrek o reprezentaciji linearnih funkcionalov: 
\begin_inset Formula $\forall\text{l.f.}\varphi:V\to F\exists!w\in V\ni:\forall v\in V:\varphi v=\left\langle v,w\right\rangle $
\end_inset


\end_layout

\begin_layout Standard
Za 
\begin_inset Formula $L:U\to V$
\end_inset

 je 
\begin_inset Formula $L^{*}:V\to U$
\end_inset

 adjungirana linearna preslika 
\begin_inset Formula $\Leftrightarrow\forall u\in U,v\in V:\left\langle Lu,v\right\rangle =\left\langle v,L^{*}u\right\rangle $
\end_inset


\end_layout

\begin_layout Standard
Za std.
 skal.
 prod.
 velja: 
\begin_inset Formula $A^{*}=\overline{A}^{T}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\left(AB\right)^{*}=B^{*}A^{*}$
\end_inset

, 
\begin_inset Formula $\left(L^{*}\right)_{B\leftarrow C}=\left(L_{C\leftarrow B}\right)^{*}$
\end_inset

, 
\begin_inset Formula $\left(\alpha A+\beta B\right)^{*}=\overline{\alpha}A^{*}+\overline{\beta}B^{*}$
\end_inset

, 
\begin_inset Formula $\left(A^{*}\right)^{*}=A$
\end_inset

, 
\begin_inset Formula $\text{Ker}L^{*}=\left(\text{Im}L\right)^{\perp}$
\end_inset

, 
\begin_inset Formula $\left(\text{Ker}L^{*}\right)^{\perp}=\text{Im}L$
\end_inset

, 
\begin_inset Formula $\text{Ker}\left(L^{*}L\right)=\text{Ker}L$
\end_inset

, 
\begin_inset Formula $\text{Im}\left(L^{*}L\right)=\text{Im}L$
\end_inset


\end_layout

\begin_layout Standard
Lastne vrednosti 
\begin_inset Formula $A^{*}$
\end_inset

 so konjugirane lastne vrednosti 
\begin_inset Formula $A$
\end_inset

.
 Dokaz: 
\begin_inset Formula $B=A-\lambda I$
\end_inset

.
 
\begin_inset Formula $B^{*}=A^{*}-\overline{\lambda}I$
\end_inset

.
 
\begin_inset Formula $\det B^{*}=\det\overline{B}^{T}=\det B=\overline{\det B}$
\end_inset

, torej 
\begin_inset Formula $\det B=0\Leftrightarrow\det B^{*}=0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\Delta_{A^{*}}$
\end_inset

 ima konjugirane koeficiente 
\begin_inset Formula $\Delta_{A}$
\end_inset

.
\end_layout

\begin_layout Paragraph
Normalne matrike 
\begin_inset Formula $A^{*}A=AA^{*}$
\end_inset


\end_layout

\begin_layout Standard
Velja: 
\begin_inset Formula $A$
\end_inset

 kvadratna, 
\begin_inset Formula $Av=\lambda v\Leftrightarrow A^{*}v=\overline{\lambda}v$
\end_inset

 (isti lastni vektorji)
\end_layout

\begin_layout Standard
\begin_inset Formula $Au=\lambda u\wedge Av=\mu v\wedge\mu\not=\lambda\Rightarrow v\perp u$
\end_inset


\end_layout

\begin_layout Standard
Je podobna diagonalni: 
\begin_inset Formula $\forall m:\text{Ker}\left(A-\lambda I\right)^{m}=\text{Ker}\left(A-\lambda I\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A=PDP^{-1}\Leftrightarrow$
\end_inset

 stolpci 
\begin_inset Formula $P$
\end_inset

 so ONB, diagonalci 
\begin_inset Formula $D$
\end_inset

 lavr, zdb 
\begin_inset Formula $P$
\end_inset

 je unitarna/ortogonalna.
\end_layout

\begin_layout Paragraph
Unitarne 
\begin_inset Formula $\mathbb{C}$
\end_inset

/ortogonalne 
\begin_inset Formula $\mathbb{R}$
\end_inset

 matrike 
\begin_inset Formula $AA^{*}=A^{*}A=I$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A$
\end_inset

 kvadratna z ON stolpci.
 
\begin_inset Formula $A$
\end_inset

 ortog.
 
\begin_inset Formula $\Rightarrow A$
\end_inset

 normalna
\end_layout

\begin_layout Standard
Lavr: let 
\begin_inset Formula $Av=\lambda v\Rightarrow\left\langle Av,Av\right\rangle =\left\langle \lambda v,\lambda v\right\rangle =\left\langle v,v\right\rangle =\lambda\overline{\lambda}\left\langle v,v\right\rangle \Rightarrow\left|\lambda\right|=1\Rightarrow\lambda=e^{i\varphi}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A=PDP^{-1},A^{*}=A^{-1}$
\end_inset

 
\end_layout

\begin_layout Paragraph
Simetrične 
\begin_inset Formula $\mathbb{R}$
\end_inset

/hermitske 
\begin_inset Formula $\mathbb{C}$
\end_inset

 matrike 
\begin_inset Formula $A=A^{*}$
\end_inset


\end_layout

\begin_layout Standard
Sebiadjungirane linearne preslikave.
\end_layout

\begin_layout Standard
Hermitska 
\begin_inset Formula $\Rightarrow$
\end_inset

 Normalna
\end_layout

\begin_layout Standard
\begin_inset Formula $Av=\lambda v=A^{*}v=\overline{\lambda}v\Rightarrow\lambda\in\mathbb{R}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A=A^{*}\Leftrightarrow\forall v:\left\langle Av,v\right\rangle \in\mathbb{R}$
\end_inset


\end_layout

\begin_layout Paragraph
Pozitivno (semi)definitne 
\begin_inset Formula $A\geq0$
\end_inset

 (
\begin_inset Formula $>$
\end_inset

 za PD)
\end_layout

\begin_layout Standard
\begin_inset Formula $A$
\end_inset

 P(S)D 
\begin_inset Formula $\Rightarrow$
\end_inset

 
\begin_inset Formula $A$
\end_inset

 sim./ortog.
 
\begin_inset Formula $\Rightarrow A$
\end_inset

 normalna
\end_layout

\begin_layout Standard
Def.: 
\begin_inset Formula $A=A^{*}\wedge\forall v:\left\langle Av,v\right\rangle \geq0$
\end_inset

 (
\begin_inset Formula $>$
\end_inset

 za PD)
\end_layout

\begin_layout Standard
Za poljubno 
\begin_inset Formula $B$
\end_inset

 je 
\begin_inset Formula $B^{*}B$
\end_inset

 PSD.
 Če ima 
\begin_inset Formula $B$
\end_inset

 LN stolpce, je 
\begin_inset Formula $B^{*}B$
\end_inset

 PD.
\end_layout

\begin_layout Standard
\begin_inset Formula $\forall\text{lavr}\lambda_{i}:A>0\Rightarrow\lambda_{i}>0$
\end_inset

, 
\begin_inset Formula $A\geq0\Rightarrow\lambda_{i}\geq0$
\end_inset

.
 Dokaz: let 
\begin_inset Formula $A\geq0,v\not=0,Av=\lambda v\Rightarrow\left\langle Av,v\right\rangle =\left\langle \lambda v,v\right\rangle =\lambda\left\langle v,v\right\rangle \geq0\wedge\left\langle v,v\right\rangle >0\Rightarrow\lambda\geq0$
\end_inset


\end_layout

\begin_layout Standard
Lavr isto kot hermitska, lave isto kot normalna, diag.
 isto kot normalna.
\end_layout

\begin_layout Standard
\begin_inset Formula $\text{A\ensuremath{\geq0}}\Rightarrow\exists B=B^{*},B\geq0\ni:B^{2}=A$
\end_inset

.
 Dokaz: let 
\begin_inset Formula $E\text{diag s koreni lavr}\geq0,A=PDP^{-1},P^{*}=P^{-1},D=\text{\text{diag z lavr}}\geq0,B=PEP^{-1}=PEP^{*}\Rightarrow B=B^{*}\Rightarrow B^{2}=PEP^{-1}PEP^{-1}=PE^{2}P^{-1}=PDP=A$
\end_inset


\end_layout

\begin_layout Standard
NTSE: 
\begin_inset Formula $A\geq0\Leftrightarrow A=A^{*}\wedge\forall\lambda\text{lavr}A:\lambda\geq0\Leftrightarrow A=PDP^{-1}\wedge P\text{ unit.}\wedge\text{diag.}D\geq0\Leftrightarrow A=A^{*}\wedge\exists\sqrt{A}\ni:\sqrt{A}^{2}=A\Leftrightarrow A=B^{*}B$
\end_inset

 (oz.
 
\begin_inset Formula $>$
\end_inset

 za PD)
\end_layout

\begin_layout Standard
\begin_inset Formula $\forall\left[\cdot,\cdot\right]:V^{2}\to F\exists M>0\ni:\forall v,u\in V:\left[v,u\right]=\left\langle Au,v\right\rangle $
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\forall A>0:\left\langle A\cdot,\cdot\right\rangle $
\end_inset

 je skalarni produkt.
\end_layout

\begin_layout Paragraph
Singularni razcep (SVD)
\end_layout

\begin_layout Standard
Singularne vrednosti 
\begin_inset Formula $A$
\end_inset

 so kvadratni koreni lastnih vrednosti 
\begin_inset Formula $A^{*}A$
\end_inset

.
\end_layout

\begin_layout Standard
Št.
 ničelnih singvr 
\begin_inset Formula $=\dim\text{Ker}\left(A^{*}A\right)=\dim\text{Ker}A$
\end_inset


\end_layout

\begin_layout Standard
Št.
 nenič.
 singvr 
\begin_inset Formula $n\times n$
\end_inset

 matrike 
\begin_inset Formula $=n-\dim\text{Ker}A=\text{rang}A$
\end_inset


\end_layout

\begin_layout Standard
Za posplošeno diagonalno matriko 
\begin_inset Formula $D$
\end_inset

 velja 
\begin_inset Formula $\forall i,j:i\not=j\Rightarrow D_{ij}=0$
\end_inset


\end_layout

\begin_layout Standard
Izred o SVD: 
\begin_inset Formula $\forall A\in M_{m\times n}\left(\mathbb{C}\right)\exists\text{unit. }Q_{1},\text{unit. }Q_{2},\text{diag. }D\ni:A=Q_{1}DQ_{2}^{-1}=Q_{1}DQ_{2}^{*}$
\end_inset

.
 Diagonalci 
\begin_inset Formula $D$
\end_inset

 so singvr 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $A^{*}=Q_{2}D^{*}Q_{1}^{*}$
\end_inset

, 
\begin_inset Formula $A^{*}A=Q_{2}D^{*}DQ_{1}^{*}\sim D^{*}D$
\end_inset

.
 Diagonalci 
\begin_inset Formula $D^{*}D$
\end_inset

 so lavr 
\begin_inset Formula $A^{*}A$
\end_inset

 in stolpci 
\begin_inset Formula $Q_{2}$
\end_inset

 so ONB lave 
\begin_inset Formula $A^{*}A$
\end_inset

.
\end_layout

\begin_layout Standard
Konstrukcija 
\begin_inset Formula $Q_{2}$
\end_inset

: ONB iz pripadajočih ONB 
\begin_inset Formula $A^{*}A$
\end_inset

.
 
\begin_inset Formula $r=\text{rang}A$
\end_inset


\end_layout

\begin_layout Standard
Konstrukcija 
\begin_inset Formula $Q_{1}$
\end_inset

: 
\begin_inset Formula $\forall i\in\left\{ 1..r\right\} :u_{i}=\frac{1}{\sigma_{i}}Av_{i}$
\end_inset

.
 
\begin_inset Formula $\left\{ u_{1},\dots,u_{r}\right\} $
\end_inset

 dopolnimo do ONB, 
\begin_inset Formula $Q_{1}=\left[\begin{array}{ccccc}
u_{1} & \cdots & u_{r} & \cdots & u_{m}\end{array}\right]$
\end_inset

 unitarna (ONB stolpci)
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{multicols}
\end_layout

\end_inset


\end_layout

\end_body
\end_document