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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\DeclareMathOperator{\Lin}{Lin}
+\DeclareMathOperator{\rang}{rang}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 0cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
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+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\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