diff options
Diffstat (limited to 'šola/la')
| -rw-r--r-- | šola/la/dn8/dokument.lyx | 305 | ||||
| -rw-r--r-- | šola/la/kolokvij4.lyx | 1068 |
2 files changed, 1285 insertions, 88 deletions
diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx index 7edbce2..c603fcc 100644 --- a/šola/la/dn8/dokument.lyx +++ b/šola/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 @@ -417,22 +420,23 @@ 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 @@ -440,20 +444,6 @@ Na predavanjih 2024-05-08 smo dokazali, \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 $A^{*}$ \end_inset @@ -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 @@ -665,6 +653,98 @@ Preizkus s programom SageMath. \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 \begin_layout Enumerate Pokaži @@ -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 @@ -1185,6 +1307,13 @@ Rokopisi, \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 \end_inset diff --git a/šola/la/kolokvij4.lyx b/šola/la/kolokvij4.lyx new file mode 100644 index 0000000..3e8a3e8 --- /dev/null +++ b/šola/la/kolokvij4.lyx @@ -0,0 +1,1068 @@ +#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}}} +\end_preamble +\use_default_options true +\begin_modules +enumitem +\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 true +\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 2cm +\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 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 |