diff options
Diffstat (limited to 'šola/la/dn8/dokument.lyx')
| -rw-r--r-- | šola/la/dn8/dokument.lyx | 305 |
1 files changed, 217 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 |