diff options
Diffstat (limited to '')
| -rw-r--r-- | šola/la/dn8/dokument.lyx | 639 | ||||
| -rw-r--r-- | šola/la/dn8/sage.png | bin | 0 -> 158070 bytes |
2 files changed, 630 insertions, 9 deletions
diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx index 3032fdd..c01e7bc 100644 --- a/šola/la/dn8/dokument.lyx +++ b/šola/la/dn8/dokument.lyx @@ -53,7 +53,7 @@ theorems-ams \spacing single \use_hyperref false \papersize default -\use_geometry false +\use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 @@ -78,12 +78,12 @@ theorems-ams \shortcut idx \color #008000 \end_index -\leftmargin 1cm -\topmargin 0cm -\rightmargin 1cm +\leftmargin 2cm +\topmargin 2cm +\rightmargin 2cm \bottommargin 2cm -\headheight 1cm -\headsep 1cm +\headheight 2cm +\headsep 2cm \footskip 1cm \secnumdepth 3 \tocdepth 3 @@ -166,7 +166,7 @@ Dokaži, da je A=\left[\begin{array}{ccc} 0 & 2 & -2\\ 0 & 1 & 0\\ -1 & 2 & -1 +-1 & 2 & -1 \end{array}\right] \] @@ -193,7 +193,12 @@ Predpostavljam polje \begin_inset Formula $V=\mathbb{R}^{3}$ \end_inset -. +, 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 + +). \begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ \end_inset @@ -211,6 +216,87 @@ Predpostavljam polje \end_inset +\begin_inset Formula +\[ +\left[\left(x,y,z\right),\left(x,y,z\right)\right]=2x^{2}-2xy+2y^{2}-2yz+z^{2}=2\left(x^{2}-xy+y^{2}\right)-2yz+z^{2}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\left(\left(x-\frac{y}{2}\right)^{2}+\frac{3y^{2}}{4}\right)-2yz+z^{2}=2\left(x-\frac{y}{2}\right)^{2}-\frac{3y^{2}}{4}-2yz+z^{2}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\left(x-\frac{y}{2}\right)^{2}+\left(\frac{\sqrt{3}y}{\sqrt{2}}-\frac{z\sqrt{2}}{\sqrt{3}}\right)^{2}+\frac{z^{2}}{3}\geq0 +\] + +\end_inset + +Sedaj poiščimo ničle. + Fiksirajmo poljubna +\begin_inset Formula $y$ +\end_inset + +, +\begin_inset Formula $z$ +\end_inset + + in uporabimo obrazec za ničle kvadratne enačbe: +\begin_inset Formula +\[ +x_{1,2}=\frac{2y\pm\sqrt{4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)}}{4} +\] + +\end_inset + +Iščemo pozitivne diskriminante. +\begin_inset Formula +\[ +4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)=-12y^{2}+16yz-8z^{2}=4 +\] + +\end_inset + +Fiksirajmo poljuben +\begin_inset Formula $z$ +\end_inset + +. + Vodilni koeficient kvadratne enačbe je negativen. + Uporabimo obrazec: +\begin_inset Formula +\[ +y_{1,2}=\frac{-16z\pm\sqrt{256z^{2}-384z^{2}=-128z^{2}}}{-24} +\] + +\end_inset + +Diskriminanta je nenegativna +\begin_inset Formula $\Leftrightarrow z=0$ +\end_inset + +. + Torej +\begin_inset Formula $z=0$ +\end_inset + +, zato +\begin_inset Formula $y=0$ +\end_inset + + in tudi +\begin_inset Formula $x=0$ +\end_inset + + glede na obrazce. + Skalarni produkt je res pozitivno definiten. \end_layout \begin_layout Enumerate @@ -218,6 +304,30 @@ Predpostavljam polje \end_inset +\begin_inset Formula +\[ +\left[\left(u,v,w\right),\left(x,y,z\right)\right]=\left[\left(x,y,z\right),\left(u,v,w\right)\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +2ux-vx-uy-2vy-wy-vz+wz=2xu-yu-xv+2yv-zv-vz+wz +\] + +\end_inset + + +\begin_inset Formula +\[ +0=0 +\] + +\end_inset + +Skalarni produkt je res simetričen. \end_layout \begin_layout Enumerate @@ -225,6 +335,313 @@ Predpostavljam polje \end_inset +\begin_inset Formula +\[ +\left[\alpha\left(\left(x_{1},y_{1},z_{1}\right)+\left(x_{2},y_{2},z_{2}\right)\right),\left(u,v,w\right)\right]= +\] + +\end_inset + + +\begin_inset Formula +\[ +=2\alpha\left(x_{1}+x_{2}\right)u-\alpha\left(y_{1}+y_{2}\right)u-\alpha\left(x_{1}+x_{2}\right)v+2\alpha\left(y_{1}+y_{2}\right)v-\alpha\left(z_{1}+z_{2}\right)v-\alpha\left(y_{1}+y_{2}\right)w+\alpha\left(z_{1}+z_{2}\right)w= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha\left(2\left(x_{1}+x_{2}\right)u-\left(y_{1}+y_{2}\right)u-\left(x_{1}+x_{2}\right)v+2\left(y_{1}+y_{2}\right)v-\left(z_{1}+z_{2}\right)v-\left(y_{1}+y_{2}\right)w+\left(z_{1}+z_{2}\right)w\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha\left(2x_{1}u+2x_{2}u-y_{1}u-y_{2}u-x_{1}v-x_{2}v+2y_{1}v+2y_{2}v-z_{1}v-z_{2}v-y_{1}w-y_{2}w+z_{1}w+z_{2}w\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\alpha\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right] +\] + +\end_inset + +Skalarni produkt je res homogen in linearen v prvem faktorju. +\end_layout + +\begin_layout Standard +Po definiciji +\begin_inset Formula $A$ +\end_inset + + normalna +\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$ +\end_inset + +. + Izračunajmo torej matriko +\begin_inset Formula $A^{*}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +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 +\begin_inset Formula $M$ +\end_inset + +, da velja +\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle=u^{*}v$ +\end_inset + +, 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 $A^{*}$ +\end_inset + + pri podani matriki +\begin_inset Formula $A$ +\end_inset + + in skalarnem produktu +\begin_inset Formula $\left[\cdot,\cdot\right]$ +\end_inset + + s pripadajočo matriko +\begin_inset Formula $M$ +\end_inset + +: +\begin_inset Formula +\[ +\left[A^{*}x,y\right]=\left[x,Ay\right]\text{, uporabimo prvo točko:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,My\right\rangle =\left\langle x,MAy\right\rangle \text{, pišimo \ensuremath{z=My}:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,z\right\rangle =\left\langle x,MAM^{-1}z\right\rangle \text{, upoštevajmo drugo točko:} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle A^{*}x,z\right\rangle =\left\langle M^{-1}A^{\square}Mx,z\right\rangle \text{, kjer je \ensuremath{A^{\square}} adjungacija \ensuremath{A} pri standardnem skalarnem produktu} +\] + +\end_inset + + +\begin_inset Formula +\[ +\Rightarrow A^{*}=M^{-1}A^{\square}M=M^{-1}\overline{A}^{T}M\overset{A\in M\left(\mathbb{R}\right)}{=}M^{-1}A^{T}M +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Potrebujemo še matriko skalarnega produkta. +\begin_inset Formula +\[ +\left\langle \left(x,y,z\right),M\left(u,v,w\right)\right\rangle =\left[\begin{array}{ccc} +x & y & z\end{array}\right]\left[\begin{array}{ccc} +m_{11} & m_{12} & m_{13}\\ +m_{21} & m_{22} & m_{23}\\ +m_{31} & m_{32} & m_{33} +\end{array}\right]\left[\begin{array}{c} +u\\ +v\\ +w +\end{array}\right]=\left[\begin{array}{ccc} +x & y & z\end{array}\right]\left[\begin{array}{c} +um_{11}+vm_{12}+wm_{13}\\ +um_{21}+vm_{22}+wm_{23}\\ +um_{31}+vm_{32}+wm_{33} +\end{array}\right]= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\begin{array}{ccccc} + & xum_{11} & xvm_{12} & xwm_{13} & +\\ ++ & yum_{21} & yvm_{22} & ywm_{23} & +\\ ++ & zum_{31} & zvm_{32} & zwm_{33} +\end{array}=\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw\text{, torej} +\] + +\end_inset + + +\begin_inset Formula +\[ +M=\left[\begin{array}{ccc} +2 & -1 & 0\\ +-1 & 2 & -1\\ +0 & -1 & 1 +\end{array}\right]\text{, njen inverz pa je }M^{-1}=\left[\begin{array}{ccc} +1 & 1 & 1\\ +1 & 2 & 2\\ +1 & 2 & 3 +\end{array}\right] +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Izračunamo +\begin_inset Formula $A^{*}$ +\end_inset + + po formuli +\begin_inset Formula $A^{*}=M^{-1}A^{T}M$ +\end_inset + + in preverimo +\begin_inset Formula $A^{*}A=AA^{*}$ +\end_inset + +: +\begin_inset Formula +\[ +A^{*}=\left[\begin{array}{ccc} +1 & 1 & 1\\ +1 & 2 & 2\\ +1 & 2 & 3 +\end{array}\right]\left[\begin{array}{ccc} +0 & 0 & -1\\ +2 & 1 & 2\\ +-2 & 0 & -1 +\end{array}\right]\left[\begin{array}{ccc} +2 & -1 & 0\\ +-1 & 2 & -1\\ +0 & -1 & 1 +\end{array}\right]=\left[\begin{array}{ccc} +-1 & 2 & -1\\ +-2 & 3 & -1\\ +-6 & 6 & -2 +\end{array}\right] +\] + +\end_inset + + +\begin_inset Formula +\[ +A^{*}A=\left[\begin{array}{ccc} +1 & -2 & 3\\ +1 & -3 & 5\\ +2 & -10 & 14 +\end{array}\right]\not=\left[\begin{array}{ccc} +8 & -6 & 2\\ +-2 & 3 & -1\\ +3 & -2 & 1 +\end{array}\right]=AA^{*} +\] + +\end_inset + + +\begin_inset Formula $A$ +\end_inset + + ni normala matrika. +\end_layout + +\begin_layout Itemize +Da preverimo pravilnost matrike +\begin_inset Formula $A^{*}$ +\end_inset + +, lahko napravimo preizkus: +\begin_inset Float figure +placement H +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset Graphics + filename sage.png + width 100col% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Preizkus s programom SageMath. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + \end_layout \end_deeper @@ -240,6 +657,78 @@ Pokaži je pozitivno semidefinitna. \end_layout +\begin_deeper +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Itemize +Definiciji: +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Formula $A:V\to V$ +\end_inset + + je normalna +\begin_inset Formula $\Leftrightarrow A^{*}A=A^{*}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $A:V\to V$ +\end_inset + + je pozitivno semidefinitna +\begin_inset Formula $\Leftrightarrow A=A^{*}\wedge\forall v\in V:\left\langle Av,v\right\rangle \geq0$ +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Itemize +\begin_inset Formula $\left(\Rightarrow\right)$ +\end_inset + + Po predpostavki velja +\begin_inset Formula $AA^{*}=A^{*}A\Rightarrow AA^{*}-A^{*}A=0$ +\end_inset + +. +\begin_inset Formula +\[ +AA^{*}-A^{*}A\overset{?}{=}\left(AA^{*}-A^{*}A\right)^{*}\Leftrightarrow0=0^{*} +\] + +\end_inset + + +\begin_inset Formula +\[ +\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle =\left\langle 0v,v\right\rangle \overset{\text{homogenost}}{=}0\left\langle v,v\right\rangle =0\geq0 +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\left(\Leftarrow\right)$ +\end_inset + +Po predpostavki velja +\begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$ +\end_inset + + +\end_layout + +\end_deeper \begin_layout Enumerate Naj bo \begin_inset Formula $w_{1}=\left(1,1,1,1\right)$ @@ -250,13 +739,107 @@ Naj bo \end_inset in -\begin_inset Formula $w_{3}=\left(6,0,2,0\right)$ +\begin_inset Formula $y=\left(6,0,2,0\right)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Enumerate +Poišči ortonormirano bazo za +\begin_inset Formula $W=\Lin\left\{ w_{1},w_{2}\right\} $ +\end_inset + + glede na standardni skalarni produkt. +\end_layout + +\begin_layout Enumerate +Izrazi +\begin_inset Formula $y$ +\end_inset + + kot vsoto vektorja iz +\begin_inset Formula $W$ +\end_inset + + in vektorja iz +\begin_inset Formula $W^{\perp}$ +\end_inset + +. +\end_layout + +\begin_layout Paragraph +Rešitev +\end_layout + +\begin_layout Enumerate +Uporabimo Gram-Schmidtov postopek in sproti normiramo bazne vektorje: +\begin_inset Formula +\[ +v_{1}=\left(1,1,1,1\right)/2=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) +\] + +\end_inset + + +\begin_inset Formula +\[ +\tilde{v_{2}}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),v_{1}\right\rangle v_{1}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\left(3,3,-1,-1\right)-2\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)=\left(2,2,-2,-2\right),\quad\quad\quad v_{2}=\tilde{v_{2}}/\left|\left|\tilde{v_{2}}\right|\right|=\text{\ensuremath{\tilde{v_{2}}/4}}=\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right) +\] + +\end_inset + +Baza za +\begin_inset Formula $W$ +\end_inset + + je +\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{-1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $ \end_inset . \end_layout \begin_layout Enumerate +Uporabimo fourierov razvoj +\begin_inset Formula $y$ +\end_inset + + po +\begin_inset Formula $B$ +\end_inset + +. +\begin_inset Formula +\[ +y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{i}+\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)= +\] + +\end_inset + + +\begin_inset Formula +\[ +=4\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)+2\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)=\left(2,2,2,2\right)+\left(1,1,-1,-1\right) +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate Poišči singularni razcep matrike \begin_inset Formula \[ @@ -509,5 +1092,43 @@ Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja. Zavedam se namreč njihovega neličnega izgleda. \end_layout +\begin_layout Standard +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8a.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/1ladn8c.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/3ladn8.jpg + +\end_inset + + +\begin_inset External + template PDFPages + filename /mnt/slu/shramba/upload/www/d/4ladn8.jpg + +\end_inset + + +\end_layout + \end_body \end_document diff --git a/šola/la/dn8/sage.png b/šola/la/dn8/sage.png Binary files differnew file mode 100644 index 0000000..1cd41f5 --- /dev/null +++ b/šola/la/dn8/sage.png |
