From 4aeb337939c65fd5c6b0c66fe7c546f2de9893df Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= <anton@sijanec.eu>
Date: Sun, 26 May 2024 22:15:19 +0200
Subject: ladn8

---
 "\305\241ola/la/dn8/dokument.lyx" | 305 +++++++++++++++++++++++++++-----------
 1 file changed, 217 insertions(+), 88 deletions(-)

(limited to 'šola/la/dn8')

diff --git "a/\305\241ola/la/dn8/dokument.lyx" "b/\305\241ola/la/dn8/dokument.lyx"
index 7edbce2..c603fcc 100644
--- "a/\305\241ola/la/dn8/dokument.lyx"
+++ "b/\305\241ola/la/dn8/dokument.lyx"
@@ -1,5 +1,5 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
 \begin_document
 \begin_header
 \save_transient_properties true
@@ -21,17 +21,18 @@
 }%
 \DeclareMathOperator{\Lin}{Lin}
 \DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
 \end_preamble
 \use_default_options true
 \begin_modules
 enumitem
 theorems-ams
 \end_modules
-\maintain_unincluded_children no
+\maintain_unincluded_children false
 \language slovene
 \language_package default
-\inputencoding auto-legacy
-\fontencoding auto
+\inputencoding auto
+\fontencoding global
 \font_roman "default" "default"
 \font_sans "default" "default"
 \font_typewriter "default" "default"
@@ -39,9 +40,7 @@ theorems-ams
 \font_default_family default
 \use_non_tex_fonts false
 \font_sc false
-\font_roman_osf false
-\font_sans_osf false
-\font_typewriter_osf false
+\font_osf false
 \font_sf_scale 100 100
 \font_tt_scale 100 100
 \use_microtype false
@@ -75,9 +74,7 @@ theorems-ams
 \suppress_date false
 \justification false
 \use_refstyle 1
-\use_formatted_ref 0
 \use_minted 0
-\use_lineno 0
 \index Index
 \shortcut idx
 \color #008000
@@ -100,16 +97,11 @@ theorems-ams
 \papercolumns 1
 \papersides 1
 \paperpagestyle default
-\tablestyle default
 \tracking_changes false
 \output_changes false
-\change_bars false
-\postpone_fragile_content false
 \html_math_output 0
 \html_css_as_file 0
 \html_be_strict false
-\docbook_table_output 0
-\docbook_mathml_prefix 1
 \end_header
 
 \begin_body
@@ -165,13 +157,11 @@ euler{e}
 \end_layout
 
 \begin_layout Enumerate
-Dokaži,
- da je 
+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
+ skalarni produkt in ugotovi, ali je
 \begin_inset Formula 
 \[
 A=\left[\begin{array}{ccc}
@@ -204,8 +194,8 @@ Predpostavljam polje
 \begin_inset Formula $V=\mathbb{R}^{3}$
 \end_inset
 
-,
- saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti je 
+, saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti
+ je 
 \begin_inset Formula $\left[\left(1,1,1+i\right),\left(1,1,1+i\right)\right]=2$
 \end_inset
 
@@ -214,8 +204,7 @@ Predpostavljam polje
 \begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$
 \end_inset
 
- je skalarni produkt,
- če zadošča naslednjim lastnostim.
+ je skalarni produkt, če zadošča naslednjim lastnostim.
  Dokažimo jih za 
 \begin_inset Formula $\left[\cdot,\cdot\right]$
 \end_inset
@@ -256,8 +245,7 @@ Sedaj poiščimo ničle.
 \begin_inset Formula $y$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $z$
 \end_inset
 
@@ -300,8 +288,7 @@ Diskriminanta je nenegativna
 \begin_inset Formula $z=0$
 \end_inset
 
-,
- zato 
+, zato 
 \begin_inset Formula $y=0$
 \end_inset
 
@@ -351,7 +338,15 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-\left[\alpha\left(\left(x_{1},y_{1},z_{1}\right)+\left(x_{2},y_{2},z_{2}\right)\right),\left(u,v,w\right)\right]=
+\left[\alpha_{1}\left(x_{1},y_{1},z_{1}\right)+\alpha_{2}\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left[\left(\alpha_{1}x_{1}+\alpha_{2}x_{2},\alpha_{1}y_{1}+\alpha_{2}y_{2},\alpha_{1}z_{1}+\alpha_{2}z_{2}\right),\left(u,v,w\right)\right]=
 \]
 
 \end_inset
@@ -359,7 +354,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=2\alpha\left(x_{1}+x_{2}\right)u-\alpha\left(y_{1}+y_{2}\right)u-\alpha\left(x_{1}+x_{2}\right)v+2\alpha\left(y_{1}+y_{2}\right)v-\alpha\left(z_{1}+z_{2}\right)v-\alpha\left(y_{1}+y_{2}\right)w+\alpha\left(z_{1}+z_{2}\right)w=
+=2\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)u-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)u-\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)v+
 \]
 
 \end_inset
@@ -367,7 +362,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2\left(x_{1}+x_{2}\right)u-\left(y_{1}+y_{2}\right)u-\left(x_{1}+x_{2}\right)v+2\left(y_{1}+y_{2}\right)v-\left(z_{1}+z_{2}\right)v-\left(y_{1}+y_{2}\right)w+\left(z_{1}+z_{2}\right)w\right)=
++2\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)v-\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)v-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)w+\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)w=
 \]
 
 \end_inset
@@ -375,7 +370,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2x_{1}u+2x_{2}u-y_{1}u-y_{2}u-x_{1}v-x_{2}v+2y_{1}v+2y_{2}v-z_{1}v-z_{2}v-y_{1}w-y_{2}w+z_{1}w+z_{2}w\right)=
+=2\alpha_{1}x_{1}u+2\alpha_{2}x_{2}u-\alpha_{1}y_{1}u-\alpha_{2}y_{2}u-\alpha_{1}x_{1}v-\alpha_{2}x_{2}v+
 \]
 
 \end_inset
@@ -383,7 +378,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)=
++2\alpha_{1}y_{1}v+2\alpha_{2}y_{2}v-\alpha_{1}z_{1}v-\alpha_{2}z_{2}v-\alpha_{1}y_{1}w-\alpha_{2}y_{2}w+\alpha_{1}z_{1}w+\alpha_{2}z_{2}w=
 \]
 
 \end_inset
@@ -391,7 +386,15 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]
+=\alpha_{1}\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha_{2}\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\alpha_{1}\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha_{2}\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]
 \]
 
 \end_inset
@@ -417,42 +420,29 @@ Po definiciji
 \end_layout
 
 \begin_layout Itemize
-Na predavanjih 2024-05-08 smo dokazali,
- da za vsak skalarni produkt 
+Na predavanjih 2024-05-08 smo dokazali, da za vsak skalarni produkt 
 \begin_inset Formula $\left[u,v\right]$
 \end_inset
 
- obstaja taka pozitivno definitna matrika 
+ obstaja taka ortogonalna (
+\begin_inset Formula $M^{*}=M^{-1}$
+\end_inset
+
+) pozitivno definitna matrika 
 \begin_inset Formula $M$
 \end_inset
 
-,
- da velja 
-\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle=u^{*}v$
+, da velja 
+\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle$
 \end_inset
 
-,
- kjer je 
+, kjer je 
 \begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
  standardni skalarni produkt.
 \end_layout
 
-\begin_layout Itemize
-Na predavanjih 2024-04-17 smo dokazali,
- da 
-\begin_inset Formula $\left[L^{*}\right]_{C\leftarrow B}=\left(\left[L\right]_{B\leftarrow C}\right)^{*}$
-\end_inset
-
-,
- torej 
-\begin_inset Formula $PLP^{-1}=\left(P^{-1}L^{*}P\right)^{*}$
-\end_inset
-
-.
-\end_layout
-
 \begin_layout Itemize
 Izpeljimo predpis za 
 \begin_inset Formula $A^{*}$
@@ -632,11 +622,9 @@ Da preverimo pravilnost matrike
 \begin_inset Formula $A^{*}$
 \end_inset
 
-,
- lahko napravimo preizkus:
+, lahko napravimo preizkus:
 \begin_inset Float figure
 placement H
-alignment document
 wide false
 sideways false
 status open
@@ -663,6 +651,98 @@ Preizkus s programom SageMath.
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+Dokazati, da 
+\begin_inset Formula $A$
+\end_inset
+
+ ni normalna, je moč še lažje.
+ Dokažemo lahko namreč, da eden izmed potrebnih pogojev za normalnost matrike
+ ni izpolnjen.
+ Na primer: 
+\begin_inset Formula $AA^{*}=A^{*}A\rightarrow A=PDP^{-1}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $P$
+\end_inset
+
+ ortogonalna in 
+\begin_inset Formula $D$
+\end_inset
+
+ diagonalna 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ lastni vektorji 
+\begin_inset Formula $A$
+\end_inset
+
+ tvorijo ortogonalno množico.
+\end_layout
+
+\begin_layout Standard
+Lastne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so (s kalkulatorjem) 
+\begin_inset Formula $\left\{ -2,1\right\} $
+\end_inset
+
+, kjer ima 1 algebrajsko večkratnost 2.
+ Lastni vektorji:
+\begin_inset Formula 
+\[
+A-\left(-2\right)I=\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+-1 & 2 & 1
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 3 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 0 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=z,y=0\Rightarrow v_{1}=\left(1,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A-1I=\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+-1 & 2 & -2
+\end{array}\right]\sim\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=2y-2z\Rightarrow v_{2}=\left(2,1,0\right),\quad v_{3}=\left(-2,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[v_{1},v_{2}\right]=\left[\left(1,0,1\right),\left(2,1,0\right)\right]=4-0-1+0-1-0+0=2\not=0\Rightarrow v_{1}\not\perp v_{2}\Rightarrow A\text{ ni normalna}
+\]
+
+\end_inset
+
+
 \end_layout
 
 \end_deeper
@@ -693,7 +773,7 @@ Definiciji:
 \end_inset
 
  je normalna 
-\begin_inset Formula $\Leftrightarrow A^{*}A=A^{*}$
+\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$
 \end_inset
 
 
@@ -746,9 +826,63 @@ Po predpostavki velja
 \begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$
 \end_inset
 
+ in 
+\begin_inset Formula $\forall v\in V:\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle \geq0$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\sled\left(AA^{*}-A^{*}A\right)=\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)\overset{\text{lastnost sledi}}{=}\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)=0
+\]
+
+\end_inset
+
+Sled 
+\begin_inset Formula $M$
+\end_inset
+
+ je vsota lastnih vrednosti 
+\begin_inset Formula $M$
+\end_inset
+
+, torej je vsota lastnih vrednosti 
+\begin_inset Formula $\left(AA^{*}-A^{*}A\right)=0$
+\end_inset
+
+.
  
-\series bold
-TODO TODO TODO XXX XXX XXX XXX XXX XXX TODO TODO TODO
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow$
+\end_inset
+
+ vse lastne vrednosti so nenegativne.
+ Iz teh dveh trditev sledi, da je vsaka lastna vrednost 
+\begin_inset Formula $AA^{*}-A^{*}A=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow AA^{*}-A^{*}A$
+\end_inset
+
+ normalna.
+ Normalne matrike je moč diagonalizirati v ortonormirani bazi:
+\begin_inset Formula 
+\[
+AA^{*}-A^{*}A=PDP^{-1}\overset{\text{diagonalci so lastne vrednosti}}{=}P0P^{-1}=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+AA^{*}=A^{*}A\Rightarrow A\text{ je normalna}
+\]
+
+\end_inset
+
+
 \end_layout
 
 \end_deeper
@@ -757,8 +891,7 @@ 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
 
@@ -852,10 +985,8 @@ Dopolnimo
 \begin_inset Formula $W^{\perp}$
 \end_inset
 
-,
- nato uporabimo Fourierov razvoj po dopolnjeni bazi.
- Bazo podprostora dopolnimo tako,
- da rešimo sistem enačb.
+, nato uporabimo Fourierov razvoj po dopolnjeni bazi.
+ Bazo podprostora dopolnimo tako, da rešimo sistem enačb.
 \begin_inset Formula 
 \[
 \left\langle \left(x_{1},y_{1},z_{1},w_{1}\right),\left(3,3,-1,-1\right)\right\rangle =0\quad\quad\quad\left\langle \left(x_{2},y_{2},z_{2},w_{2}\right),\left(1,1,1,1\right)\right\rangle =0
@@ -943,8 +1074,7 @@ Iščemo
 \begin_inset Formula $U$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $\Sigma$
 \end_inset
 
@@ -952,8 +1082,7 @@ Iščemo
 \begin_inset Formula $V$
 \end_inset
 
-,
- da velja 
+, da velja 
 \begin_inset Formula $A=U\Sigma V^{*}$
 \end_inset
 
@@ -978,18 +1107,15 @@ Diagonalci
 \begin_inset Formula $A^{*}A$
 \end_inset
 
-,
- torej 
+, 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
 
@@ -1054,8 +1180,8 @@ Stolpci
 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)
+0 & 0 & -4
+\end{array}\right]\Rightarrow x=z=0\Rightarrow v_{1}=\left(0,1,0\right)
 \]
 
 \end_inset
@@ -1066,8 +1192,8 @@ A^{*}A-4I=\left[\begin{array}{ccc}
 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)
+0 & 0 & -1
+\end{array}\right]\Rightarrow y=z=0\Rightarrow v_{2}=\left(1,0,0\right)
 \]
 
 \end_inset
@@ -1114,8 +1240,7 @@ Stolpci
 \begin_inset Formula $v_{\rang A+1},\dots,v_{m}$
 \end_inset
 
- najdemo tako,
- da dopolnimo 
+ najdemo tako, da dopolnimo 
 \begin_inset Formula $v_{1},\dots,v_{\rang A}$
 \end_inset
 
@@ -1136,8 +1261,7 @@ U=\left[\begin{array}{cccc}
 \end_layout
 
 \begin_layout Itemize
-Dobljene matrike zmnožimo,
- s čimer potrdimo veljavnost singularnega razcepa:
+Dobljene matrike zmnožimo, s čimer potrdimo veljavnost singularnega razcepa:
 \begin_inset Formula 
 \[
 U\Sigma V^{*}=\left[\begin{array}{cccc}
@@ -1169,9 +1293,7 @@ U\Sigma V^{*}=\left[\begin{array}{cccc}
 
 \end_deeper
 \begin_layout Standard
-Rokopisi,
- ki sledijo,
- naj služijo le kot dokaz samostojnega reševanja.
+Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja.
  Zavedam se namreč njihovega neličnega izgleda.
 \end_layout
 
@@ -1183,6 +1305,13 @@ Rokopisi,
 \end_inset
 
 
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8aq.jpg
+
+\end_inset
+
+
 \begin_inset External
 	template PDFPages
 	filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg
-- 
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