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| <!DOCTYPE html> | ||
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| <html lang="pl" data-content_root="../../../"> | ||
| <head> | ||
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| <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" /> | ||
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| <title>Endomorfizmy — Matematyka - dokumentacja</title> | ||
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| <li class="nav-item nav-item-this"><a href="">Endomorfizmy</a></li> | ||
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| <div class="document"> | ||
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| <section id="endomorfizmy"> | ||
| <h1>Endomorfizmy<a class="headerlink" href="#endomorfizmy" title="Link to this heading">¶</a></h1> | ||
| <div class="admonition-endomorfizm admonition"> | ||
| <p class="admonition-title">Endomorfizm</p> | ||
| <p><strong>Odwzorowanie liniowe</strong> jednego zbioru na ten sam zbiór.</p> | ||
| </div> | ||
| <div class="admonition note"> | ||
| <p class="admonition-title">Informacja</p> | ||
| <p>Zbiór wszystkich endomorfizmów ma strukturę przestrzeni wektorowej</p> | ||
| </div> | ||
| <div class="admonition-phi-niezmiennicza admonition"> | ||
| <p class="admonition-title"><span class="math notranslate nohighlight">\(\phi\)</span> niezmiennicza</p> | ||
| <p>jeżeli <code class="docutils literal notranslate"><span class="pre">u</span></code> jest podzbiorem V, dla <span class="math notranslate nohighlight">\(x \in u\)</span> <span class="math notranslate nohighlight">\(\phi(x) \in u\)</span></p> | ||
| </div> | ||
| <div class="admonition-wartosc-wlasna admonition"> | ||
| <p class="admonition-title">wartość włąsna</p> | ||
| <p>Liczba <span class="math notranslate nohighlight">\(\lambda\)</span> jest wartością własną jężeli istnieje niezerowy wektor <code class="docutils literal notranslate"><span class="pre">v</span></code>, któ©y spełnia | ||
| równość <span class="math notranslate nohighlight">\(\phi(v) = \lambda v\)</span></p> | ||
| <div class="admonition note"> | ||
| <p class="admonition-title">Informacja</p> | ||
| <p>Zbió© wszystkich wartości własnych nazywa się spektrum (lub widmem) operatora</p> | ||
| </div> | ||
| </div> | ||
| <div class="admonition-operator-rzutowania admonition"> | ||
| <p class="admonition-title">Operator rzutowania</p> | ||
| <p>Określea wszystkie wektory które po zrzutowaniu na oś OX nie zmieni kierunku (lub stanie się punktem)</p> | ||
| <div class="math notranslate nohighlight"> | ||
| \[\begin{split} | ||
| spec = ? \\ | ||
| \phi(0, V_y) = (0, v_y) \Rightarrow \lambda = 1 \\ | ||
| \phi(0, v_y) = (0, 0) \Rightarrow \lambda = 0 \\ | ||
| spec = \left \{0, 1\right\} \\ | ||
| \end{split}\]</div> | ||
| </div> | ||
| <section id="szukanie-wartosci-wlasnych"> | ||
| <h2>Szukanie wartości własnych<a class="headerlink" href="#szukanie-wartosci-wlasnych" title="Link to this heading">¶</a></h2> | ||
| <div class="math notranslate nohighlight"> | ||
| \[ | ||
| x_\phi(t) = det(A-tI) | ||
| \]</div> | ||
| <div class="admonition important"> | ||
| <p class="admonition-title">Ważne</p> | ||
| <p>Nie wszystkie pierwiastki ww. równania są pierwiastkami charakterystycznymi. | ||
| Należy pamiętać, że <span class="math notranslate nohighlight">\(t = \lambda \in K\)</span>, gdzie <code class="docutils literal notranslate"><span class="pre">K</span></code> to ciało nad którym pracujemy</p> | ||
| </div> | ||
| <div class="admonition note"> | ||
| <p class="admonition-title">Informacja</p> | ||
| <p>Wielomiany włąsne nie zależą od bazy</p> | ||
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| <li><a class="reference internal" href="#">Endomorfizmy</a><ul> | ||
| <li><a class="reference internal" href="#szukanie-wartosci-wlasnych">Szukanie wartości własnych</a></li> | ||
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