https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&feed=atom&action=history 01NUM1:Kapitola2 - Historie editací 2024-03-29T10:56:22Z Historie editací této stránky MediaWiki 1.25.2 https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7537&oldid=prev Kubuondr: label pro lemma v konvergence geometrické posloupnosti. 2017-01-30T16:14:44Z <p>label pro lemma v konvergence geometrické posloupnosti.</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 1. 2017, 16:14</td> </tr><tr><td colspan="2" class="diff-lineno" id="L452" >Řádka 452:</td> <td colspan="2" class="diff-lineno">Řádka 452:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \max\limits_{\lVert \vec y \rVert_2 = 1} \sum_{i = 1}^n \lvert \lambda_i \rvert \lvert y_i \rvert^2 = \lambda_k = \rho ( \matice {A^* A} ) = \lVert \matice A \rVert_2^2 \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \max\limits_{\lVert \vec y \rVert_2 = 1} \sum_{i = 1}^n \lvert \lambda_i \rvert \lvert y_i \rvert^2 = \lambda_k = \rho ( \matice {A^* A} ) = \lVert \matice A \rVert_2^2 \]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Je-li \(\matice A\) hermitovská, platí \(\matice {A^*A} = \matice A^2\) a \(\lVert \matice A \rVert _2 = \sqrt{\rho(\matice A^2)} = \rho(\matice A)\). (Rovnost plyne z <del class="diffchange diffchange-inline">bezprostředně následujícího lemmatu</del>)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Je-li \(\matice A\) hermitovská, platí \(\matice {A^*A} = \matice A^2\) a \(\lVert \matice A \rVert _2 = \sqrt{\rho(\matice A^2)} = \rho(\matice A)\). (Rovnost plyne z <ins class="diffchange diffchange-inline">\ref{MocninaJordan}</ins>)</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\\ Je-li \(\matice A\) unitární, pak \(\lVert \matice A \rVert _2 = 1\) \qedhere</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\\ Je-li \(\matice A\) unitární, pak \(\lVert \matice A \rVert _2 = 1\) \qedhere</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark*}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L462" >Řádka 462:</td> <td colspan="2" class="diff-lineno">Řádka 462:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{lemma*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{lemma*}</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\label{MocninaJordan}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Nechť \( \matice J \in \mathbbm C^{n, n} \) je Jordanovou maticí z rozkladu \ref{Jordan}. Potom platí</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Nechť \( \matice J \in \mathbbm C^{n, n} \) je Jordanovou maticí z rozkladu \ref{Jordan}. Potom platí</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ (\matice J^k)_{ij} =</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ (\matice J^k)_{ij} =</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7532&oldid=prev Kubuondr v 30. 1. 2017, 07:57 2017-01-30T07:57:49Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 1. 2017, 07:57</td> </tr><tr><td colspan="2" class="diff-lineno" id="L259" >Řádka 259:</td> <td colspan="2" class="diff-lineno">Řádka 259:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; 0 &amp; \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; 0 &amp; \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Naprosto stejným postupem pokračujeme dále, až dojdeme k matici obsahující na diagonále vlastní čísla matice \(\matice A\) a případné další nenulové prvky nad diagonálou (ty tam kvůli jedničkám v maticích \(\matice H_2\) <del class="diffchange diffchange-inline">a </del>a dalších zůstanou). Tu označíme jako matici \(\matice R\). Dále označíme</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Naprosto stejným postupem pokračujeme dále, až dojdeme k matici obsahující na diagonále vlastní čísla matice \(\matice A\) a případné další nenulové prvky nad diagonálou (ty tam kvůli jedničkám v maticích \(\matice H_2\) a dalších zůstanou). Tu označíme jako matici \(\matice R\). Dále označíme</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \matice U = \prod_{i = 0}^{n - 1} \matice H_{n - i} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \matice U = \prod_{i = 0}^{n - 1} \matice H_{n - i} \]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Protože jsou všechny matice \(\matice H(\vec w_k)\) Householderovy reflekční matice, jsou podle \ref{HouseholderHermUnit} unitární. Součin unitárních matic je unitární matice (důkaz na dva řádky je trivální), tedy celá matice \(\matice U\) je unitární. Matice \(\matice U^{-1}\) bude mít tvar \(\matice H_1 \matice H_2 ... \matice H_n \). To už je ekvivalentní s tvrzením věty: \[\matice U^* \matice A \matice U = \matice R \Leftrightarrow \matice A = \matice U^* \matice R \matice U\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Protože jsou všechny matice \(\matice H(\vec w_k)\) Householderovy reflekční matice, jsou podle \ref{HouseholderHermUnit} unitární. Součin unitárních matic je unitární matice (důkaz na dva řádky je trivální), tedy celá matice \(\matice U\) je unitární. Matice \(\matice U^{-1}\) bude mít tvar \(\matice H_1 \matice H_2 ... \matice H_n \). To už je ekvivalentní s tvrzením věty: \[\matice U^* \matice A \matice U = \matice R \Leftrightarrow \matice A = \matice U^* \matice R \matice U\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7530&oldid=prev Kubuondr: oprava důkazu Schurovy věty: nad diagonálou nejsou obecně nuly (u nediagonalizovatelných matic) 2017-01-29T16:58:34Z <p>oprava důkazu Schurovy věty: nad diagonálou nejsou obecně nuly (u nediagonalizovatelných matic)</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 29. 1. 2017, 16:58</td> </tr><tr><td colspan="2" class="diff-lineno" id="L253" >Řádka 253:</td> <td colspan="2" class="diff-lineno">Řádka 253:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]\[=</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]\[=</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{pmatrix} &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{pmatrix} &#160;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\lambda_1 &amp; <del class="diffchange diffchange-inline">0 </del>&amp; \cdots \\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\lambda_1 &amp; <ins class="diffchange diffchange-inline">r_1 </ins>&amp; \cdots \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; \lambda_2 &amp; \cdots \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; \lambda_2 &amp; \cdots \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\vdots &amp; 0 &amp; \multirow{3}{*}{ \huge {\( \matice A'' \)} }\\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\vdots &amp; 0 &amp; \multirow{3}{*}{ \huge {\( \matice A'' \)} }\\</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L259" >Řádka 259:</td> <td colspan="2" class="diff-lineno">Řádka 259:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; 0 &amp; \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>0 &amp; 0 &amp; \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Naprosto stejným postupem pokračujeme dále, až dojdeme k matici obsahující <del class="diffchange diffchange-inline">pouze </del>vlastní čísla matice \(\matice A\). Tu označíme jako matici \(\matice R\). Dále označíme</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Naprosto stejným postupem pokračujeme dále, až dojdeme k matici obsahující <ins class="diffchange diffchange-inline">na diagonále </ins>vlastní čísla matice \(\matice A\<ins class="diffchange diffchange-inline">) a případné další nenulové prvky nad diagonálou (ty tam kvůli jedničkám v maticích \(\matice H_2\) a a dalších zůstanou</ins>). Tu označíme jako matici \(\matice R\). Dále označíme</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \matice U = \prod_{i = 0}^{n - 1} \matice H_{n - i} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \matice U = \prod_{i = 0}^{n - 1} \matice H_{n - i} \]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Protože jsou všechny matice \(\matice H(\vec w_k)\) Householderovy reflekční matice, jsou podle \ref{HouseholderHermUnit} unitární. Součin unitárních matic je unitární matice (důkaz na dva řádky je trivální), tedy celá matice \(\matice U\) je unitární. Matice \(\matice U^{-1}\) bude mít tvar \(\matice H_1 \matice H_2 ... \matice H_n \). To už je ekvivalentní s tvrzením věty: \[\matice U^* \matice A \matice U = \matice R \Leftrightarrow \matice A = \matice U^* \matice R \matice U\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Protože jsou všechny matice \(\matice H(\vec w_k)\) Householderovy reflekční matice, jsou podle \ref{HouseholderHermUnit} unitární. Součin unitárních matic je unitární matice (důkaz na dva řádky je trivální), tedy celá matice \(\matice U\) je unitární. Matice \(\matice U^{-1}\) bude mít tvar \(\matice H_1 \matice H_2 ... \matice H_n \). To už je ekvivalentní s tvrzením věty: \[\matice U^* \matice A \matice U = \matice R \Leftrightarrow \matice A = \matice U^* \matice R \matice U\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7473&oldid=prev Kubuondr v 25. 1. 2017, 14:08 2017-01-25T14:08:09Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 14:08</td> </tr><tr><td colspan="2" class="diff-lineno" id="L182" >Řádka 182:</td> <td colspan="2" class="diff-lineno">Řádka 182:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">(</del>Protože je Householedora matice unitární, musíme normovat, jinak by totiž \(\matice H(\vec w)\vec x\) nemohl být jednotkový vektor<del class="diffchange diffchange-inline">)</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Protože je Householedora matice unitární, musíme <ins class="diffchange diffchange-inline">vektor \(\vec x\) </ins>normovat, jinak by totiž \(\matice H(\vec w)\vec x\) nemohl být jednotkový vektor<ins class="diffchange diffchange-inline">.</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7472&oldid=prev Kubuondr: zdůvodnění sqrt(\rho(A^2))=\rho(A) 2017-01-25T13:06:05Z <p>zdůvodnění sqrt(\rho(A^2))=\rho(A)</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 13:06</td> </tr><tr><td colspan="2" class="diff-lineno" id="L452" >Řádka 452:</td> <td colspan="2" class="diff-lineno">Řádka 452:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \max\limits_{\lVert \vec y \rVert_2 = 1} \sum_{i = 1}^n \lvert \lambda_i \rvert \lvert y_i \rvert^2 = \lambda_k = \rho ( \matice {A^* A} ) = \lVert \matice A \rVert_2^2 \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \max\limits_{\lVert \vec y \rVert_2 = 1} \sum_{i = 1}^n \lvert \lambda_i \rvert \lvert y_i \rvert^2 = \lambda_k = \rho ( \matice {A^* A} ) = \lVert \matice A \rVert_2^2 \]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Je-li \(\matice A\) hermitovská, platí \(\matice {A^*A} = \matice A^2\) a \(\lVert \matice A \rVert _2 = \sqrt{\rho(\matice A^2)} = \rho(\matice A)\). &#160;</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Je-li \(\matice A\) hermitovská, platí \(\matice {A^*A} = \matice A^2\) a \(\lVert \matice A \rVert _2 = \sqrt{\rho(\matice A^2)} = \rho(\matice A)\). <ins class="diffchange diffchange-inline">(Rovnost plyne z bezprostředně následujícího lemmatu)</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\\ Je-li \(\matice A\) unitární, pak \(\lVert \matice A \rVert _2 = 1\) \qedhere</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\\ Je-li \(\matice A\) unitární, pak \(\lVert \matice A \rVert _2 = 1\) \qedhere</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark*}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7471&oldid=prev Kubuondr v 25. 1. 2017, 12:16 2017-01-25T12:16:22Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 12:16</td> </tr><tr><td colspan="2" class="diff-lineno" id="L329" >Řádka 329:</td> <td colspan="2" class="diff-lineno">Řádka 329:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}, \;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}, \;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\forall k \in \hat p \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\forall k \in \hat p \]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Počet bloků příslušejících k \(\lambda\) je roven \(\nu_g(\lambda)\) a součet řádů těchto bloků je \(\<del class="diffchange diffchange-inline">nu_y</del>(\lambda)\).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Počet bloků příslušejících k \(\lambda\) je roven \(\nu_g(\lambda)\) a součet řádů těchto bloků je \(\<ins class="diffchange diffchange-inline">nu_a</ins>(\lambda)\).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Matice \( \matice J \) je až na pořadí bloků dána jednoznačně.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Matice \( \matice J \) je až na pořadí bloků dána jednoznačně.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}\renewcommand{\qedsymbol}{}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}\renewcommand{\qedsymbol}{}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7469&oldid=prev Kubuondr: dodatečné informace v Jordanově větě. 2017-01-25T12:12:25Z <p>dodatečné informace v Jordanově větě.</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 12:12</td> </tr><tr><td colspan="2" class="diff-lineno" id="L329" >Řádka 329:</td> <td colspan="2" class="diff-lineno">Řádka 329:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}, \;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}, \;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\forall k \in \hat p \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\forall k \in \hat p \]</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Počet bloků příslušejících k \(\lambda\) je roven \(\nu_g(\lambda)\) a součet řádů těchto bloků je \(\nu_y(\lambda)\).</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Matice \( \matice J \) je až na pořadí bloků dána jednoznačně.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Matice \( \matice J \) je až na pořadí bloků dána jednoznačně.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}\renewcommand{\qedsymbol}{}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}\renewcommand{\qedsymbol}{}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7468&oldid=prev Kubuondr: doplněno rovnítko, 2017-01-25T11:46:45Z <p>doplněno rovnítko,</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 11:46</td> </tr><tr><td colspan="2" class="diff-lineno" id="L296" >Řádka 296:</td> <td colspan="2" class="diff-lineno">Řádka 296:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item Ukážeme, že \(\matice R\) je normální, pak podle \ref{NormalniTrojuhelnikDiagonalni} bude také diagonální.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item Ukážeme, že \(\matice R\) je normální, pak podle \ref{NormalniTrojuhelnikDiagonalni} bude také diagonální.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A = \matice {U^*RU} \Rightarrow \matice {UA} = \matice {RU} \Rightarrow \matice {UAU^*} = \matice R\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A = \matice {U^*RU} \Rightarrow \matice {UA} = \matice {RU} \Rightarrow \matice {UAU^*} = \matice R\]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[\matice R^* = \matice {(UAU^*)^*} = \matice {(AU^*)^*U^*} \matice {UA^*U^*} \]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[\matice R^* = \matice {(UAU^*)^*} = \matice {(AU^*)^*U^*} <ins class="diffchange diffchange-inline">= </ins>\matice {UA^*U^*} \]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice {RR^*} = \matice {UA}\underbrace{\matice{U^*U}}_{\matice I}\matice {A^*U^*} = \matice {UAA^*U^*} = \matice {UA^*AU^*} = \matice {UA^*U^*UAU^*} = \matice{R^*R}\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice {RR^*} = \matice {UA}\underbrace{\matice{U^*U}}_{\matice I}\matice {A^*U^*} = \matice {UAA^*U^*} = \matice {UA^*AU^*} = \matice {UA^*U^*UAU^*} = \matice{R^*R}\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\(\Rightarrow \matice R \) je normální \(\Rightarrow \matice R\) je diagonální.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\(\Rightarrow \matice R \) je normální \(\Rightarrow \matice R\) je diagonální.</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7465&oldid=prev Kubuondr v 25. 1. 2017, 10:00 2017-01-25T10:00:17Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 10:00</td> </tr><tr><td colspan="2" class="diff-lineno" id="L182" >Řádka 182:</td> <td colspan="2" class="diff-lineno">Řádka 182:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(Protože je Householedora matice unitární, musíme normovat, jinak by totiž \(\matice H(\vec w)\vec x\) nemohl být <del class="diffchange diffchange-inline">jenotkový </del>vektor)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(Protože je Householedora matice unitární, musíme normovat, jinak by totiž \(\matice H(\vec w)\vec x\) nemohl být <ins class="diffchange diffchange-inline">jednotkový </ins>vektor)</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola2&diff=7464&oldid=prev Kubuondr: zdůvodnění normování vektoru v důkazu 2.37 2017-01-25T09:59:20Z <p>zdůvodnění normování vektoru v důkazu 2.37</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 25. 1. 2017, 09:59</td> </tr><tr><td colspan="2" class="diff-lineno" id="L182" >Řádka 182:</td> <td colspan="2" class="diff-lineno">Řádka 182:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a pokud vezmeme \(\vec x\) jako normovaný:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec w = \frac{\vec e^{(1)}-\vec x}{\lVert \vec e^{(1)} - \vec x \rVert_2} \]</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(Protože je Householedora matice unitární, musíme normovat, jinak by totiž \(\matice H(\vec w)\vec x\) nemohl být jenotkový vektor)</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Z volby \( \vec w \) pak plyne:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\matice A \matice H (\vec w)\vec e^{(1)} = \matice A \vec x = \lambda \vec x\]</div></td></tr> </table> Kubuondr