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author | Anton Luka Šijanec <anton@sijanec.eu> | 2024-05-18 10:10:13 +0200 |
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committer | Anton Luka Šijanec <anton@sijanec.eu> | 2024-05-18 10:10:13 +0200 |
commit | 8b1a5e554d07aafea80207c440d0909d23f61ad0 (patch) | |
tree | 7039c2546669f852d2f9621f96601c0912e355e8 /šola/la/dn8 | |
parent | searphp fix (diff) | |
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-rw-r--r-- | šola/la/dn8/dokument.lyx | 513 |
1 files changed, 513 insertions, 0 deletions
diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx new file mode 100644 index 0000000..3032fdd --- /dev/null +++ b/šola/la/dn8/dokument.lyx @@ -0,0 +1,513 @@ +#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 +\papercolumns 1 +\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 |