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    <identity>
      <id>be8ed228-7b9f-4d88-9521-f6039e697cd5</id>
      <revision>117</revision>
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    <calculation>
      <precision>4</precision>
      <exponentialThreshold>5</exponentialThreshold>
      <fractions>decimal</fractions>
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      <margins left="39" right="39" top="39" bottom="39" />
      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
      <footer alignment="Center" color="#a9a9a9">&amp;[PAGENUM] / &amp;[COUNT]</footer>
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  <region id="0" left="18" top="18" width="241" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Rechnen mit Matrizen und Vektoren</p>
    </text>
    <text lang="eng">
      <p bold="true">Matrix and vector operations</p>
    </text>
  </region>
  <region id="1" left="540" top="18" width="177" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Chapter Matrices.sm">Vektoren und Matrizen</p>
    </text>
    <text lang="eng">
      <p href="Chapter Matrices.sm">Vectors and Matrices</p>
    </text>
  </region>
  <region id="2" left="540" top="36" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="3" left="540" top="54" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="4" left="18" top="90" width="266" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p>Let Maxima handle determinants.</p>
    </text>
  </region>
  <region id="5" left="18" top="117" width="683" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand" style="string">takeover</e>
        <e type="function" preserve="true" args="1">MaximaControl</e>
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      <result action="numeric">
        <e type="operand" style="string">int(), diff(), sum(), det() and lim() now use Maxima</e>
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    </math>
  </region>
  <region id="6" left="18" top="162" width="210" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Addition und Subtraktion</p>
    </text>
    <text lang="eng">
      <p underline="true">Addition and subtraction</p>
    </text>
  </region>
  <region id="7" left="36" top="198" width="398" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Addition und Subtraktion gleich großer Matrizen</p>
    </text>
  </region>
  <region id="8" left="36" top="234" width="173" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">3</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">12</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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      <result action="numeric">
        <e type="operand">4</e>
        <e type="operand">8</e>
        <e type="operand">12</e>
        <e type="operand">16</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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    </math>
  </region>
  <region id="9" left="225" top="234" width="175" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">3</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">12</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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      <result action="numeric">
        <e type="operand">2</e>
        <e type="operator" args="1">-</e>
        <e type="operand">4</e>
        <e type="operator" args="1">-</e>
        <e type="operand">6</e>
        <e type="operator" args="1">-</e>
        <e type="operand">8</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="10" left="36" top="288" width="415" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Elementweise Addition oder Subtraktion einer Zahl</p>
    </text>
  </region>
  <region id="11" left="36" top="315" width="124" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">5</e>
        <e type="operator" args="2">+</e>
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      <result action="numeric">
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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    </math>
  </region>
  <region id="12" left="225" top="315" width="150" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">10</e>
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        <e type="operand">9</e>
        <e type="operator" args="1">-</e>
        <e type="operand">8</e>
        <e type="operator" args="1">-</e>
        <e type="operand">7</e>
        <e type="operator" args="1">-</e>
        <e type="operand">6</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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    </math>
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  <region id="13" left="18" top="378" width="224" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Multiplikation mit einer Zahl</p>
    </text>
    <text lang="eng">
      <p underline="true">Multiplication by a number</p>
    </text>
  </region>
  <region id="14" left="36" top="414" width="126" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">3</e>
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      <result action="numeric">
        <e type="operand">3</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">12</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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    </math>
  </region>
  <region id="15" left="171" top="414" width="134" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">*</e>
      </input>
      <result action="numeric">
        <e type="operand">5</e>
        <e type="operand">10</e>
        <e type="operand">15</e>
        <e type="operand">20</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="16" left="36" top="468" width="189" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">3</e>
        <e type="operator" args="2">/</e>
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      <result action="numeric">
        <e type="operand">0.3333</e>
        <e type="operand">0.6667</e>
        <e type="operand">1</e>
        <e type="operand">1.3333</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="17" left="252" top="468" width="204" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">1</e>
        <e type="operand">3</e>
        <e type="operator" args="2">/</e>
        <e type="operator" args="2">*</e>
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      <result action="numeric">
        <e type="operand">0.3333</e>
        <e type="operand">0.6667</e>
        <e type="operand">1</e>
        <e type="operand">1.3333</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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    </math>
  </region>
  <region id="18" left="468" top="468" width="245" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Division durch eine Zahl 
entspricht der Multiplikation
mit derem Kehrwert.</p>
    </text>
    <text lang="eng">
      <p>Division by a number is 
equivalent to multiplication 
by it's inverse.</p>
    </text>
  </region>
  <region id="19" left="18" top="540" width="123" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Transposition </p>
    </text>
  </region>
  <region id="20" left="36" top="567" width="131" height="71" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="8">mat</e>
        <e type="function" preserve="true" args="1">transpose</e>
      </input>
      <result action="numeric">
        <e type="operand">1</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">2</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="8">mat</e>
      </result>
    </math>
  </region>
  <region id="21" left="252" top="567" width="153" height="32" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="22" left="450" top="576" width="32" height="33" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">#</e>
        <e type="function" preserve="true" args="1">transpose</e>
      </input>
    </math>
  </region>
  <region id="23" left="414" top="585" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>-&gt;</p>
    </text>
  </region>
  <region id="24" left="252" top="594" width="42" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="25" left="306" top="603" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">trans</e>
      </input>
    </math>
  </region>
  <region id="26" left="360" top="603" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="27" left="18" top="648" width="184" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Matrixmultiplikation </p>
    </text>
    <text lang="eng">
      <p underline="true">Matrix multiplication</p>
    </text>
  </region>
  <region id="28" left="36" top="684" width="199" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="8">mat</e>
        <e type="operand">1</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">2</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="8">mat</e>
        <e type="operator" args="2">*</e>
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      <result action="numeric">
        <e type="operand">5</e>
        <e type="operand">11</e>
        <e type="operand">17</e>
        <e type="operand">11</e>
        <e type="operand">25</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">39</e>
        <e type="operand">61</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
      </result>
    </math>
  </region>
  <region id="29" left="243" top="684" width="159" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="8">mat</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">*</e>
      </input>
      <result action="numeric">
        <e type="operand">7</e>
        <e type="operand">10</e>
        <e type="operand">15</e>
        <e type="operand">22</e>
        <e type="operand">23</e>
        <e type="operand">34</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="8">mat</e>
      </result>
    </math>
  </region>
  <region id="30" left="414" top="684" width="306" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die Spaltenzahl der ersten Matrix
muss gleich der Zeilenzahl der 
zweiten Matrix sein.</p>
    </text>
    <text lang="eng">
      <p>The number of columns of the left 
matrix must match the number of rows
in the right matrix.</p>
    </text>
  </region>
  <region id="31" left="18" top="756" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Determinante</p>
    </text>
    <text lang="eng">
      <p underline="true">Determinant</p>
    </text>
  </region>
  <region id="32" left="36" top="792" width="68" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operand">d</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="33" left="261" top="792" width="154" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="34" left="135" top="801" width="110" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="symbolic">
        <e type="operand">a</e>
        <e type="operand">d</e>
        <e type="operator" args="2">*</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
      </result>
    </math>
  </region>
  <region id="35" left="423" top="819" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>-&gt;</p>
    </text>
  </region>
  <region id="36" left="261" top="828" width="42" height="35" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="37" left="459" top="828" width="26" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">#</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
    </math>
  </region>
  <region id="38" left="315" top="837" width="37" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">det</e>
      </input>
    </math>
  </region>
  <region id="39" left="351" top="837" width="37" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Tab</p>
    </text>
  </region>
  <region id="40" left="36" top="846" width="68" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="operand">0</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="41" left="135" top="855" width="68" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="numeric">
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
      </result>
    </math>
  </region>
  <region id="42" left="18" top="900" width="142" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Rang einer Matrix</p>
    </text>
    <text lang="eng">
      <p underline="true">Rank of a matrix</p>
    </text>
  </region>
  <region id="43" left="36" top="927" width="68" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="operand">0</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="44" left="144" top="936" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="numeric">
        <e type="operand">2</e>
      </result>
    </math>
  </region>
  <region id="45" left="36" top="972" width="98" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">D</e>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">2</e>
        <e type="operand">a</e>
        <e type="operator" args="2">*</e>
        <e type="operand">2</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="46" left="144" top="981" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">D</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="symbolic">
        <e type="operand">1</e>
      </result>
    </math>
  </region>
  <region id="47" left="18" top="1017" width="697" height="72" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die numerische Berechnung des Rangs ist nicht robust, da die Zwischenergebnisse für 
die interne Zahlendarstellung zu groß werden. Im Einzelfall kann die symbolische 
Auswertung noch sinnvolle Werte liefern, stößt aber auch an Grenzen der 
Zahlendarstellung. </p>
    </text>
    <text lang="eng">
      <p>The numerical computation of the rank in SMath is not robust, because the
intermediate results hit the limit of number representation. In special cases, the 
symbolic evaluation may provide useful results</p>
    </text>
  </region>
  <region id="48" left="36" top="1089" width="196" height="188" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">4</e>
        <e type="operand">8</e>
        <e type="operand">8</e>
        <e type="operand">2</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="operand">5</e>
        <e type="operand">3</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">4</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">4</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">0</e>
        <e type="operand">0</e>
        <e type="operand">3</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">6</e>
        <e type="operand">5</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">2</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">6</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">5</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">2</e>
        <e type="operand">4</e>
        <e type="operand">8</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">6</e>
        <e type="operand">5</e>
        <e type="operand">9</e>
        <e type="operand">3</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">10</e>
        <e type="operand">10</e>
        <e type="function" preserve="true" args="102">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="49" left="243" top="1089" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math error="43">
      <description active="true" position="Top" lang="ger">
        <p>Numerische Optimierung</p>
      </description>
      <description active="true" position="Top" lang="eng">
        <p>numeric optimization</p>
      </description>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="numeric">
        <e type="operand">#</e>
      </result>
    </math>
  </region>
  <region id="50" left="243" top="1134" width="135" height="34" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="numeric">
        <e type="operand">7.8489</e>
        <e type="operand">10</e>
        <e type="operand">8</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">*</e>
      </result>
    </math>
  </region>
  <region id="51" left="243" top="1188" width="106" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <description active="true" position="Top" lang="ger">
        <p>symbolische Optimierung</p>
      </description>
      <description active="true" position="Top" lang="eng">
        <p>symbolic optimization</p>
      </description>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="symbolic">
        <e type="operand">10</e>
      </result>
    </math>
  </region>
  <region id="52" left="243" top="1242" width="125" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="symbolic">
        <e type="operand">784894425</e>
      </result>
    </math>
  </region>
  <region id="53" left="36" top="1278" width="167" height="28" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
        <e type="function" preserve="true" args="1">Maxima</e>
      </input>
      <result action="numeric">
        <e type="operand">10</e>
      </result>
    </math>
  </region>
  <region id="54" left="18" top="1314" width="127" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Inverse Matrix </p>
    </text>
    <text lang="eng">
      <p underline="true">Matrix inverse</p>
    </text>
  </region>
  <region id="55" left="36" top="1350" width="180" height="78" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">2</e>
        <e type="operand">a</e>
        <e type="operator" args="2">*</e>
        <e type="operand">3</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
      </input>
      <result action="symbolic">
        <e type="operand">3</e>
        <e type="operand">a</e>
        <e type="operator" args="2">/</e>
        <e type="operand">1</e>
        <e type="operand">a</e>
        <e type="operator" args="2">/</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="operand">b</e>
        <e type="operator" args="2">/</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operand">b</e>
        <e type="operator" args="2">/</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="56" left="36" top="1431" width="647" height="188" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="3" exponentialThreshold="6">
      <input>
        <e type="operand">A</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
      </input>
      <result action="numeric">
        <e type="operand">0.1</e>
        <e type="operand">0.145</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.119</e>
        <e type="operand">0.079</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.025</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.075</e>
        <e type="operand">0.052</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.146</e>
        <e type="operand">0.016</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.146</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.043</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.063</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.038</e>
        <e type="operand">0.052</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.058</e>
        <e type="operand">0.018</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.052</e>
        <e type="operand">0.09</e>
        <e type="operand">0.022</e>
        <e type="operand">0.064</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.008</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.03</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.076</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.003</e>
        <e type="operand">0.016</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.077</e>
        <e type="operand">0.021</e>
        <e type="operand">0.005</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.074</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.097</e>
        <e type="operand">0.01</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.025</e>
        <e type="operand">0.182</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.035</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.211</e>
        <e type="operand">0.032</e>
        <e type="operand">0.042</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.016</e>
        <e type="operand">0.081</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.047</e>
        <e type="operand">0.013</e>
        <e type="operand">0.003</e>
        <e type="operand">0.032</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.018</e>
        <e type="operand">0.074</e>
        <e type="operand">0.016</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.024</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.108</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.049</e>
        <e type="operand">0.051</e>
        <e type="operand">0.11</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.1</e>
        <e type="operand">0.15</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.031</e>
        <e type="operand">0.055</e>
        <e type="operand">0.024</e>
        <e type="operand">0.02</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.045</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.042</e>
        <e type="operand">0.09</e>
        <e type="operand">0.065</e>
        <e type="operand">0.076</e>
        <e type="operand">0.169</e>
        <e type="operand">0.05</e>
        <e type="operand">0.288</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.138</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.136</e>
        <e type="operand">0.008</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.099</e>
        <e type="operand">0.037</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.044</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.034</e>
        <e type="operand">0.066</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.096</e>
        <e type="operand">0.105</e>
        <e type="operand">0.077</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.049</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.046</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.015</e>
        <e type="operand">0.076</e>
        <e type="operand">0.028</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0</e>
        <e type="operand">0.094</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.033</e>
        <e type="operand">0.156</e>
        <e type="operand">0.119</e>
        <e type="operand">0.068</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.064</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.082</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.009</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.063</e>
        <e type="operand">0.073</e>
        <e type="operand">0.247</e>
        <e type="operand">0.046</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.304</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.052</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.079</e>
        <e type="operand">0.034</e>
        <e type="operand">0.004</e>
        <e type="operand">0.043</e>
        <e type="operator" args="1">-</e>
        <e type="operand">10</e>
        <e type="operand">10</e>
        <e type="function" preserve="true" args="102">mat</e>
      </result>
    </math>
  </region>
  <region id="57" left="18" top="1638" width="258" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Minor und Algebraisches Kompement</p>
    </text>
    <text lang="eng">
      <p underline="true">Minor and algebraic complement</p>
    </text>
  </region>
  <region id="58" left="126" top="1665" width="204" height="64" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">11</e>
        <e type="operand">17</e>
        <e type="operand">11</e>
        <e type="operand">25</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">39</e>
        <e type="operand">61</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">vminor</e>
      </input>
      <result action="numeric">
        <e type="operand">11</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">61</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
      </result>
    </math>
  </region>
  <region id="59" left="342" top="1674" width="356" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Durch Streichen der i-ten Zeile und j-ten 
Spalte entstehende Untermatrix von M</p>
    </text>
    <text lang="eng">
      <p>submatrix with row i and column j deleted.</p>
    </text>
  </region>
  <region id="60" left="36" top="1683" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>vminor()</p>
    </text>
  </region>
  <region id="61" left="252" top="1728" width="100" height="46" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">11</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">61</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="numeric">
        <e type="operand">8</e>
      </result>
    </math>
  </region>
  <region id="62" left="126" top="1782" width="163" height="64" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">11</e>
        <e type="operand">17</e>
        <e type="operand">11</e>
        <e type="operand">25</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">39</e>
        <e type="operand">61</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">minor</e>
      </input>
      <result action="numeric">
        <e type="operand">8</e>
      </result>
    </math>
  </region>
  <region id="63" left="342" top="1782" width="332" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Minor. Das ist die Determinante der durch 
Streichung der Zeile i und der Spalte j 
entstehenden Untermatrix. </p>
    </text>
    <text lang="eng">
      <p>Minor. This is the i,j sub-determinant.</p>
    </text>
  </region>
  <region id="64" left="36" top="1800" width="70" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>minor()</p>
    </text>
  </region>
  <region id="65" left="126" top="1854" width="172" height="64" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">11</e>
        <e type="operand">17</e>
        <e type="operand">11</e>
        <e type="operand">25</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">39</e>
        <e type="operand">61</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">alg</e>
      </input>
      <result action="numeric">
        <e type="operand">8</e>
        <e type="operator" args="1">-</e>
      </result>
    </math>
  </region>
  <region id="66" left="342" top="1854" width="344" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Algebraisches Komplement. Das ist der mit 
(-1)^(i+j)multiplizierte Minor der Matrix.</p>
    </text>
    <text lang="eng">
      <p>Algebraic complement. This is the 
i,j subdeterminant (minor) multiplied by 
(-1)^(i+j).</p>
    </text>
  </region>
  <region id="67" left="36" top="1872" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>alg()</p>
    </text>
  </region>
  <region id="68" left="36" top="1926" width="505" height="24" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Beachten Sie, dass vminor() und minor() identische Operatorform haben.</p>
    </text>
    <text lang="eng">
      <p>Note that minor() and vminor() have identical operator form.</p>
    </text>
  </region>
  <region id="69" left="18" top="1962" width="150" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Spur einer Matrix</p>
    </text>
    <text lang="eng">
      <p underline="true">Trace of a matrix</p>
    </text>
  </region>
  <region id="70" left="36" top="1998" width="154" height="64" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">5</e>
        <e type="operand">11</e>
        <e type="operand">17</e>
        <e type="operand">11</e>
        <e type="operand">25</e>
        <e type="operand">39</e>
        <e type="operand">17</e>
        <e type="operand">39</e>
        <e type="operand">61</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
        <e type="function" preserve="true" args="1">tr</e>
      </input>
      <result action="numeric">
        <e type="operand">91</e>
      </result>
    </math>
  </region>
  <region id="71" left="243" top="1998" width="242" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Summe der Diagonalelemente</p>
    </text>
    <text lang="eng">
      <p>Sum of the diagonal elements</p>
    </text>
  </region>
  <region id="72" left="18" top="2070" width="210" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Matrix- und Vektornormen</p>
    </text>
  </region>
  <region id="73" left="36" top="2106" width="68" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="8">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="74" left="135" top="2106" width="52" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="5">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="75" left="36" top="2178" width="127" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Euklidische Norm</p>
    </text>
    <text lang="eng">
      <p underline="true">Euclidean norm</p>
    </text>
  </region>
  <region id="76" left="306" top="2178" width="209" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Referenzimplementierung </p>
    </text>
    <text lang="eng">
      <p>Reference implementation</p>
    </text>
  </region>
  <region id="77" left="288" top="2205" width="189" height="43" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">M#</e>
        <e type="function" args="1">norm.e</e>
        <e type="operand">M#</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="operand">M#</e>
        <e type="operator" args="2">*</e>
        <e type="function" preserve="true" args="1">tr</e>
        <e type="function" preserve="true" args="1">sqrt</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="78" left="54" top="2214" width="131" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">norme</e>
      </input>
      <result action="numeric">
        <e type="operand">9.54</e>
      </result>
    </math>
  </region>
  <region id="79" left="567" top="2214" width="131" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" args="1">norm.e</e>
      </input>
      <result action="numeric">
        <e type="operand">9.54</e>
      </result>
    </math>
  </region>
  <region id="80" left="54" top="2241" width="143" height="35" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" preserve="true" args="1">norme</e>
      </input>
      <result action="numeric">
        <e type="operand">9.54</e>
      </result>
    </math>
  </region>
  <region id="81" left="567" top="2241" width="143" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" args="1">norm.e</e>
      </input>
      <result action="numeric">
        <e type="operand">9.54</e>
      </result>
    </math>
  </region>
  <region id="82" left="54" top="2286" width="131" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" preserve="true" args="1">norme</e>
      </input>
      <result action="numeric">
        <e type="operand">3.74</e>
      </result>
    </math>
  </region>
  <region id="83" left="567" top="2286" width="131" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" args="1">norm.e</e>
      </input>
      <result action="numeric">
        <e type="operand">3.74</e>
      </result>
    </math>
  </region>
  <region id="84" left="36" top="2313" width="151" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Spaltensummennorm </p>
    </text>
    <text lang="eng">
      <p underline="true">L1 (row sum) norm</p>
    </text>
  </region>
  <region id="85" left="765" top="2313" width="741" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand" style="string">surrender</e>
        <e type="function" preserve="true" args="1">MaximaControl</e>
      </input>
      <result action="numeric">
        <e type="operand" style="string">int(), diff(), sum(), det() and lim() now are native SMath</e>
      </result>
    </math>
  </region>
  <region id="86" left="54" top="2340" width="114" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">norm1</e>
      </input>
      <result action="numeric">
        <e type="operand">12</e>
      </result>
    </math>
  </region>
  <region id="87" left="279" top="2340" width="281" height="105" color="#000000" bgColor="#ffffff" fontSize="10">
    <math optimize="2" decimalPlaces="2">
      <input>
        <e type="operand">M#</e>
        <e type="function" args="1">norm.1</e>
        <e type="operand">j</e>
        <e type="operand">1</e>
        <e type="operand">M#</e>
        <e type="function" preserve="true" args="1">cols</e>
        <e type="function" preserve="true" args="2">range</e>
        <e type="operand">a#</e>
        <e type="operand">j</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="operand">M#</e>
        <e type="operand">i</e>
        <e type="operand">j</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="function" preserve="true" args="1">abs</e>
        <e type="operand">i</e>
        <e type="operand">1</e>
        <e type="operand">M#</e>
        <e type="function" preserve="true" args="1">rows</e>
        <e type="function" preserve="true" args="4">sum</e>
        <e type="operator" args="2">:</e>
        <e type="function" preserve="true" args="3">for</e>
        <e type="operand">a#</e>
        <e type="function" preserve="true" args="1">max</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">line</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="88" left="567" top="2340" width="114" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" args="1">norm.1</e>
      </input>
      <result action="numeric">
        <e type="operand">12</e>
      </result>
    </math>
  </region>
  <region id="89" left="54" top="2367" width="126" height="35" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" preserve="true" args="1">norm1</e>
      </input>
      <result action="numeric">
        <e type="operand">11</e>
      </result>
    </math>
  </region>
  <region id="90" left="567" top="2367" width="126" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" args="1">norm.1</e>
      </input>
      <result action="numeric">
        <e type="operand">11</e>
      </result>
    </math>
  </region>
  <region id="91" left="54" top="2403" width="106" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" preserve="true" args="1">norm1</e>
      </input>
      <result action="numeric">
        <e type="operand">6</e>
      </result>
    </math>
  </region>
  <region id="92" left="567" top="2403" width="106" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" args="1">norm.1</e>
      </input>
      <result action="numeric">
        <e type="operand">6</e>
      </result>
    </math>
  </region>
  <region id="93" left="36" top="2448" width="226" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Zeilensummennorm </p>
    </text>
    <text lang="eng">
      <p underline="true">Infinity norm (column sum)</p>
    </text>
  </region>
  <region id="94" left="54" top="2475" width="114" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">normi</e>
      </input>
      <result action="numeric">
        <e type="operand">11</e>
      </result>
    </math>
  </region>
  <region id="95" left="279" top="2475" width="281" height="105" color="#000000" bgColor="#ffffff" fontSize="10">
    <math optimize="2" decimalPlaces="2">
      <input>
        <e type="operand">M#</e>
        <e type="function" args="1">norm.∞</e>
        <e type="operand">i</e>
        <e type="operand">1</e>
        <e type="operand">M#</e>
        <e type="function" preserve="true" args="1">rows</e>
        <e type="function" preserve="true" args="2">range</e>
        <e type="operand">a#</e>
        <e type="operand">i</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="operand">M#</e>
        <e type="operand">i</e>
        <e type="operand">j</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="function" preserve="true" args="1">abs</e>
        <e type="operand">j</e>
        <e type="operand">1</e>
        <e type="operand">M#</e>
        <e type="function" preserve="true" args="1">cols</e>
        <e type="function" preserve="true" args="4">sum</e>
        <e type="operator" args="2">:</e>
        <e type="function" preserve="true" args="3">for</e>
        <e type="operand">a#</e>
        <e type="function" preserve="true" args="1">max</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">line</e>
        <e type="operator" args="2">:</e>
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    </math>
  </region>
  <region id="96" left="567" top="2475" width="114" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" args="1">norm.∞</e>
      </input>
      <result action="numeric">
        <e type="operand">11</e>
      </result>
    </math>
  </region>
  <region id="97" left="54" top="2502" width="126" height="35" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" preserve="true" args="1">normi</e>
      </input>
      <result action="numeric">
        <e type="operand">12</e>
      </result>
    </math>
  </region>
  <region id="98" left="567" top="2502" width="126" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="function" args="1">norm.∞</e>
      </input>
      <result action="numeric">
        <e type="operand">12</e>
      </result>
    </math>
  </region>
  <region id="99" left="54" top="2538" width="106" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" preserve="true" args="1">normi</e>
      </input>
      <result action="numeric">
        <e type="operand">3</e>
      </result>
    </math>
  </region>
  <region id="100" left="567" top="2538" width="106" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="2">
      <input>
        <e type="operand">b</e>
        <e type="function" args="1">norm.∞</e>
      </input>
      <result action="numeric">
        <e type="operand">3</e>
      </result>
    </math>
  </region>
  <region id="101" left="765" top="2565" width="683" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand" style="string">takeover</e>
        <e type="function" preserve="true" args="1">MaximaControl</e>
      </input>
      <result action="numeric">
        <e type="operand" style="string">int(), diff(), sum(), det() and lim() now use Maxima</e>
      </result>
    </math>
  </region>
  <region id="102" left="18" top="2592" width="217" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Skalarprodukt von Vektoren</p>
    </text>
    <text lang="eng">
      <p underline="true">Scalar product of vectors</p>
    </text>
  </region>
  <region id="103" left="36" top="2628" width="61" height="86" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">a.1</e>
        <e type="operand">a.2</e>
        <e type="operand">a.3</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="5">mat</e>
        <e type="operator" args="2">:</e>
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    </math>
  </region>
  <region id="104" left="117" top="2628" width="61" height="86" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">b</e>
        <e type="operand">b.1</e>
        <e type="operand">b.2</e>
        <e type="operand">b.3</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="5">mat</e>
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    </math>
  </region>
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    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
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      <result action="symbolic">
        <e type="operand">a.1</e>
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        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">+</e>
        <e type="operand">a.3</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">+</e>
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    </math>
  </region>
  <region id="106" left="216" top="2655" width="173" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">a</e>
        <e type="operator" args="2">*</e>
      </input>
      <result action="symbolic">
        <e type="operand">a.1</e>
        <e type="operand">2</e>
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        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">+</e>
        <e type="operand">a.3</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">+</e>
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    </math>
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  <region id="107" left="216" top="2691" width="170" height="41" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
      </input>
      <result action="symbolic">
        <e type="operand">a.1</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operand">a.2</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">+</e>
        <e type="operand">a.3</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">+</e>
      </result>
    </math>
  </region>
  <region id="108" left="36" top="2736" width="670" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Das Skalarprodukt kann auch als Matrixprodukt ausgedrückt werden. Dabei entsteht
dann eine 1x1-Matrix</p>
    </text>
  </region>
  <region id="109" left="36" top="2781" width="230" height="38" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
      </input>
      <result action="symbolic">
        <e type="operand">a.1</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.2</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">+</e>
        <e type="operand">a.3</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">+</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="3">mat</e>
      </result>
    </math>
  </region>
  <region id="110" left="18" top="2826" width="119" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Kreuzprodukt</p>
    </text>
    <text lang="eng">
      <p underline="true">Cross product</p>
    </text>
  </region>
  <region id="111" left="36" top="2862" width="170" height="86" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operator" args="2">†</e>
      </input>
      <result action="symbolic">
        <e type="operand">a.2</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.3</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operand">a.3</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.1</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operand">a.1</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.2</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="5">mat</e>
      </result>
    </math>
  </region>
  <region id="112" left="18" top="2952" width="127" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Dyadisches (Tensor-)Produkt</p>
    </text>
    <text lang="eng">
      <p underline="true">Dyadic product</p>
    </text>
  </region>
  <region id="113" left="36" top="2988" width="220" height="86" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="function" preserve="true" args="1">transpose</e>
        <e type="operator" args="2">*</e>
      </input>
      <result action="symbolic">
        <e type="operand">a.1</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.1</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.1</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.2</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.2</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.2</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.3</e>
        <e type="operand">b.1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.3</e>
        <e type="operand">b.2</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a.3</e>
        <e type="operand">b.3</e>
        <e type="operator" args="2">*</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="function" preserve="true" args="11">mat</e>
      </result>
    </math>
  </region>
  <region id="114" left="18" top="3096" width="243" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Eigenwerte und Eigenvektoren</p>
    </text>
    <text lang="eng">
      <p underline="true">Eigenvalues and Eigenvectors</p>
    </text>
  </region>
  <region id="115" left="36" top="3132" width="93" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
        <e type="operand">100</e>
        <e type="operand">50</e>
        <e type="operand">50</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="116" top="3177" color="#000000" bgColor="#ffffff">
    <area collapsed="true">
      <title lang="eng">
        <p>DotNumerics</p>
      </title>
    </area>
    <region id="117" left="36" top="3195" width="147" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p>Mit dem Plugin</p>
      </text>
      <text lang="eng">
        <p>Using the plugin </p>
      </text>
    </region>
    <region id="118" left="198" top="3195" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p href="Plugin DotNumerics.sm">DotNumerics</p>
      </text>
    </region>
    <region id="119" left="36" top="3222" width="220" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
      <math optimize="2">
        <input>
          <e type="operand">λ</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">dn_LinAlgEigenvalues</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="120" left="270" top="3222" width="121" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">λ</e>
        </input>
        <result action="numeric">
          <e type="operand">120.7107</e>
          <e type="operand">20.7107</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="4">mat</e>
        </result>
      </math>
    </region>
    <region id="121" left="468" top="3231" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p>Eigenwerte</p>
      </text>
      <text lang="eng">
        <p>eigenvalues</p>
      </text>
    </region>
    <region id="122" left="36" top="3276" width="228" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
      <math optimize="2">
        <input>
          <e type="operand">Φ</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">dn_LinAlgEigenvectors</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="123" left="288" top="3276" width="169" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
        </input>
        <result action="numeric">
          <e type="operand">0.9239</e>
          <e type="operand">0.3827</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.3827</e>
          <e type="operand">0.9239</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="124" left="468" top="3285" width="111" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p>Eigenvektoren </p>
      </text>
      <text lang="eng">
        <p>eigenvectors</p>
      </text>
    </region>
    <region id="125" left="36" top="3321" width="324" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="operand">M</e>
          <e type="operator" args="2">*</e>
          <e type="operand">Φ</e>
          <e type="operator" args="2">*</e>
        </input>
        <result action="numeric">
          <e type="operand">120.7107</e>
          <e type="operand">5.9863</e>
          <e type="operand">10</e>
          <e type="operand">15</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="1">-</e>
          <e type="operand">1.1318</e>
          <e type="operand">10</e>
          <e type="operand">14</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="1">-</e>
          <e type="operand">20.7107</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="126" left="36" top="3393" width="267" height="53" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
          <e type="operand">λ</e>
          <e type="function" preserve="true" args="1">diag</e>
          <e type="operator" args="2">*</e>
          <e type="operand">Φ</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="operator" args="2">*</e>
        </input>
        <result action="numeric">
          <e type="operand">100</e>
          <e type="operand">50</e>
          <e type="operand">50</e>
          <e type="operand">5.392</e>
          <e type="operand">10</e>
          <e type="operand">15</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="127" left="468" top="3393" width="201" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p>Spektraldarstellung </p>
      </text>
      <text lang="eng">
        <p>spectral representation</p>
      </text>
    </region>
    <region id="128" left="36" top="3465" width="196" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="operand">Φ</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">col</e>
          <e type="operator" args="2">*</e>
        </input>
        <result action="numeric">
          <e type="operand">111.5221</e>
          <e type="operand">46.194</e>
          <e type="operand">2</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="4">mat</e>
        </result>
      </math>
    </region>
    <region id="129" left="243" top="3465" width="210" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">col</e>
          <e type="operand">λ</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operator" args="2">*</e>
        </input>
        <result action="numeric">
          <e type="operand">111.5221</e>
          <e type="operand">46.194</e>
          <e type="operand">2</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="4">mat</e>
        </result>
      </math>
    </region>
    <region id="130" top="3501" color="#000000" bgColor="#ffffff">
      <area terminator="true" />
    </region>
  </region>
  <region id="131" top="3519" color="#000000" bgColor="#ffffff">
    <area collapsed="true">
      <title lang="eng">
        <p>Maxima</p>
      </title>
    </area>
    <region id="132" left="36" top="3537" width="147" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p>Mit dem Plugin</p>
      </text>
      <text lang="eng">
        <p>Using the plugin </p>
      </text>
    </region>
    <region id="133" left="180" top="3537" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="ger">
        <p href="Plugin Maxima.sm">Maxima</p>
      </text>
    </region>
    <region id="134" left="36" top="3573" width="374" height="48" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvalues</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="symbolic">
          <e type="operand">50</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">50</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="135" left="36" top="3627" width="396" height="48" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="symbolic">
          <e type="operand">50</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">50</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="136" left="432" top="3636" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Eigenvalues</e>
        </input>
      </math>
    </region>
    <region id="137" left="432" top="3654" width="111" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Multiplicity</e>
        </input>
      </math>
    </region>
    <region id="138" left="36" top="3672" width="330" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="symbolic">
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="4">mat</e>
        </result>
      </math>
    </region>
    <region id="139" left="432" top="3681" width="152" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="eng">
        <p>first eigenvector</p>
      </text>
    </region>
    <region id="140" left="36" top="3717" width="322" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="symbolic">
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="1">sqrt</e>
          <e type="operator" args="2">+</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="4">mat</e>
        </result>
      </math>
    </region>
    <region id="141" left="432" top="3726" width="160" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="eng">
        <p>second eigenvector</p>
      </text>
    </region>
    <region id="142" left="36" top="3762" width="153" height="62" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">i</e>
          <e type="operator" args="2">+</e>
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">i</e>
          <e type="operator" args="2">-</e>
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">3</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="11">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="143" left="207" top="3762" width="318" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvalues</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="none">
          <e type="operand">i</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="operator" args="2">-</e>
          <e type="operand">i</e>
          <e type="operand">2</e>
          <e type="operator" args="2">-</e>
          <e type="operand">1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="8">mat</e>
        </result>
      </math>
    </region>
    <region id="144" left="207" top="3807" width="294" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="numeric">
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="5">mat</e>
        </result>
      </math>
    </region>
    <region id="145" left="207" top="3852" width="294" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="numeric">
          <e type="operand">1</e>
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="5">mat</e>
        </result>
      </math>
    </region>
    <region id="146" left="207" top="3897" width="294" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="function" args="1">eigenvectors</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
        </input>
        <result action="numeric">
          <e type="operand">0</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" preserve="true" args="5">mat</e>
        </result>
      </math>
    </region>
    <region id="147" top="3942" color="#000000" bgColor="#ffffff">
      <area collapsed="true">
        <title lang="eng">
          <p>Limited translation</p>
        </title>
      </area>
      <region id="148" left="36" top="3960" width="631" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
        <text lang="eng">
          <p>The complete list structure returned by eigenvectors() is not yet correctly 
translated.</p>
        </text>
      </region>
      <region id="149" left="36" top="4014" width="225" height="28" color="#000000" bgColor="#ffffff" fontSize="10">
        <math error="9">
          <input>
            <e type="operand">M</e>
            <e type="function" args="1">eigenvectors</e>
            <e type="function" preserve="true" args="1">Maxima</e>
          </input>
          <result action="numeric">
            <e type="operand">#</e>
          </result>
        </math>
      </region>
      <region id="150" left="279" top="4014" width="700" height="115" color="#000000" bgColor="#ffffff" fontSize="10">
        <math>
          <input>
            <e type="operand">#</e>
            <e type="function" preserve="true" args="1">MaximaLog</e>
          </input>
          <result action="numeric">
            <e type="operand" style="string">
Request: eigenvectors(matrix([-2+%i,0,0],[0,-(2+%i),0],[0,0,-1]));
Answer: 
(%s68) [[[-%i-2,%i-2,-1],[1,1,1]],[[[0,1,0]],[[1,0,0]],[[0,0,1]]]]
(%i69) 
Received bytes: 75
SMath get: [[[-i-2;i-2;-1];[1;1;1]];[[[0;1;0]];[[1;0;0]];[[0;0;1]]]]</e>
          </result>
        </math>
      </region>
      <region id="151" top="4122" color="#000000" bgColor="#ffffff">
        <area terminator="true" />
      </region>
    </region>
    <region id="152" left="36" top="4149" width="382" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="eng">
        <p>Numeric procedure for real symmetric matrices</p>
      </text>
    </region>
    <region id="153" left="36" top="4176" width="127" height="80" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">4</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">6</e>
          <e type="operand">4</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">3</e>
          <e type="operand">8</e>
          <e type="operand">2</e>
          <e type="operand">6</e>
          <e type="operand">8</e>
          <e type="operand">4</e>
          <e type="operand">4</e>
          <e type="operand">4</e>
          <e type="function" preserve="true" args="18">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="154" left="189" top="4176" width="438" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
      <text lang="eng">
        <p>Expression for eigenvectors() is too long, therefore
the numeric procedure of Maxima needs to be used.</p>
      </text>
    </region>
    <region id="155" left="36" top="4257" width="262" height="36" color="#000000" bgColor="#ffffff" fontSize="10">
      <math optimize="2">
        <input>
          <e type="operand">λ</e>
          <e type="operand">M</e>
          <e type="function" args="1">eigens_by_jacobi</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="156" left="324" top="4266" width="293" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">λ</e>
        </input>
        <result action="numeric">
          <e type="operand">1.952</e>
          <e type="operator" args="1">-</e>
          <e type="operand">7.8257</e>
          <e type="operator" args="1">-</e>
          <e type="operand">6.3272</e>
          <e type="operand">13.4505</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" preserve="true" args="6">mat</e>
        </result>
      </math>
    </region>
    <region id="157" left="36" top="4302" width="262" height="36" color="#000000" bgColor="#ffffff" fontSize="10">
      <math optimize="2">
        <input>
          <e type="operand">Φ</e>
          <e type="operand">M</e>
          <e type="function" args="1">eigens_by_jacobi</e>
          <e type="operand">2</e>
          <e type="function" preserve="true" args="2">el</e>
          <e type="function" preserve="true" args="1">Maxima</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="158" left="324" top="4302" width="301" height="80" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
        </input>
        <result action="numeric">
          <e type="operand">0.8416</e>
          <e type="operand">0.0037</e>
          <e type="operand">0.4651</e>
          <e type="operand">0.2746</e>
          <e type="operand">0.3446</e>
          <e type="operand">0.5103</e>
          <e type="operand">0.7569</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.2188</e>
          <e type="operand">0.4076</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.5693</e>
          <e type="operand">0.374</e>
          <e type="operand">0.6082</e>
          <e type="operand">0.0823</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.6446</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.2662</e>
          <e type="operator" args="1">-</e>
          <e type="operand">0.7119</e>
          <e type="operand">4</e>
          <e type="operand">4</e>
          <e type="function" preserve="true" args="18">mat</e>
        </result>
      </math>
    </region>
    <region id="159" left="36" top="4329" width="236" height="80" color="#000000" bgColor="#ffffff" fontSize="10">
      <math>
        <input>
          <e type="operand">Φ</e>
          <e type="operand">λ</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="function" preserve="true" args="1">diag</e>
          <e type="operator" args="2">*</e>
          <e type="operand">Φ</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="operator" args="2">*</e>
        </input>
        <result action="numeric">
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">4</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">2</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">6</e>
          <e type="operand">4</e>
          <e type="operand">2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">3</e>
          <e type="operand">8</e>
          <e type="operand">2</e>
          <e type="operand">6</e>
          <e type="operand">8</e>
          <e type="operand">4</e>
          <e type="operand">4</e>
          <e type="operand">4</e>
          <e type="function" preserve="true" args="18">mat</e>
        </result>
      </math>
    </region>
    <region id="160" top="4410" color="#000000" bgColor="#ffffff">
      <area terminator="true" />
    </region>
  </region>
  <region id="161" left="18" top="4446" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Plugin Maxima.sm">Maxima</p>
    </text>
  </region>
  <region id="162" left="18" top="4464" width="177" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Chapter Matrices.sm">Vektoren und Matrizen</p>
    </text>
    <text lang="eng">
      <p href="Chapter Matrices.sm">Vectors and Matrices</p>
    </text>
  </region>
  <region id="163" left="18" top="4482" width="103" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="164" left="18" top="4500" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="165" left="18" top="4536" width="450" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p italic="true">$Author: mkraska $
$Date: 2013-12-01 23:13:16 +0100 (So, 01. Dez 2013) $ </p>
    </text>
  </region>
</regions>