Пример. Матричные функции нормы и определителя

Функции > Векторы и матрицы > Характеристики массива > Пример. Матричные функции нормы и определителя

Пример. Матричные функции нормы и определителя

1. Определите квадратную матрицу.

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2. Задайте для элементов матрицы имена базовых переменных.

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        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
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        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
        </id>
        <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
          <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
        </id>
      </matrix>
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">M</id>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
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3. Используйте функцию norm1, чтобы найти норму L1 матрицы M.

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Либо найдите норму L1, вычислив максимум абсолютных сумм столбцов матрицы M.

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            </id>
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            </id>
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        </apply>
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          </id>
        </apply>
      </apply>
    </define>
  </math>
</region>

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          </id>
        </apply>
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    </define>
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4. Используйте функцию norm2, чтобы найти норму L2 матрицы M.

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Либо используйте функцию svds, чтобы найти самое большое абсолютное сингулярное значение матрицы M.

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Функция svds возвращает вектор сортированных сингулярных значений; таким образом, верхнее значение является самым большим сингулярным значением матрицы M.

5. Используйте функцию norme, чтобы найти евклидову норму матрицы M.

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Или же вручную вычислите квадратный корень суммы абсолютных квадратов матрицы M.

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                        <apply xml:lang="ru">
                          <pow xml:lang="ru" />
                          <apply xml:lang="ru">
                            <absval xml:lang="ru" />
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                <apply xml:lang="ru">
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                  <apply xml:lang="ru">
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                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
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6. Используйте функцию normi, чтобы найти бесконечную норму матрицы M.

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Либо используйте функцию максимума, чтобы вручную вычислить максимум абсолютных сумм строк матрицы M.

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            <absval xml:lang="ru" />
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
            </id>
          </apply>
          <apply xml:lang="ru">
            <absval xml:lang="ru" />
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
            </id>
          </apply>
        </apply>
        <apply xml:lang="ru">
          <absval xml:lang="ru" />
          <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
            <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
          </id>
        </apply>
      </apply>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
</region>

Нажать для копирования этого выражения

<region id="ID0EF1KMB" actualWidth="155.11333333333334" actualHeight="27.791200000000003" top="1209.6000000000001" left="38.400000000000006" xml:lang="ru" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="70" xml:lang="ru">
    <define xml:lang="ru" xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
        <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">ARS<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
      </id>
      <apply xml:lang="ru">
        <plus xml:lang="ru" />
        <apply xml:lang="ru">
          <plus xml:lang="ru" />
          <apply xml:lang="ru">
            <absval xml:lang="ru" />
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
            </id>
          </apply>
          <apply xml:lang="ru">
            <absval xml:lang="ru" />
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
            </id>
          </apply>
        </apply>
        <apply xml:lang="ru">
          <absval xml:lang="ru" />
          <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
            <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
          </id>
        </apply>
      </apply>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
</region>

Нажать для копирования этого выражения

<region id="ID0EX1KMB" actualWidth="252.23666666666671" actualHeight="27.791200000000003" top="1248.0000000000002" left="38.400000000000006" xml:lang="ru" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="72" xml:lang="ru">
    <define xml:lang="ru" xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">
        <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">N<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">i</Subscript></Span>
      </id>
      <eval xml:lang="ru">
        <apply xml:lang="ru">
          <id labels="FUNCTION" xml:space="preserve" xml:lang="ru" label-is-contextual="true">max</id>
          <sequence xml:lang="ru">
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">ARS<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
            </id>
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">ARS<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
            </id>
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">ARS<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
            </id>
          </sequence>
        </apply>
        <unitOverride xml:lang="ru">
          <placeholder xml:lang="ru" />
        </unitOverride>
      </eval>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
</region>

7. Используйте функцию det, чтобы найти определитель матрицы M.

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Либо вручную вычислите определитель матрицы M.

Нажать для копирования этого выражения

<region id="ID0E54KMB" actualWidth="490.83" actualHeight="27.791200000000003" top="1401.6000000000001" left="38.400000000000006" xml:lang="ru" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="76" xml:lang="ru">
    <define xml:lang="ru" xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">DET</id>
      <eval xml:lang="ru">
        <apply xml:lang="ru">
          <plus xml:lang="ru" />
          <apply xml:lang="ru">
            <minus xml:lang="ru" />
            <apply xml:lang="ru">
              <mult xml:lang="ru" />
              <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
              </id>
              <parens xml:lang="ru">
                <apply xml:lang="ru">
                  <minus xml:lang="ru" />
                  <apply xml:lang="ru">
                    <mult xml:lang="ru" />
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
                    </id>
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
                    </id>
                  </apply>
                  <apply xml:lang="ru">
                    <mult xml:lang="ru" />
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
                    </id>
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
                    </id>
                  </apply>
                </apply>
              </parens>
            </apply>
            <apply xml:lang="ru">
              <mult xml:lang="ru" />
              <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
              </id>
              <parens xml:lang="ru">
                <apply xml:lang="ru">
                  <minus xml:lang="ru" />
                  <apply xml:lang="ru">
                    <mult xml:lang="ru" />
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                    </id>
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
                    </id>
                  </apply>
                  <apply xml:lang="ru">
                    <mult xml:lang="ru" />
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
                    </id>
                    <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                      <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                    </id>
                  </apply>
                </apply>
              </parens>
            </apply>
          </apply>
          <apply xml:lang="ru">
            <mult xml:lang="ru" />
            <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
              <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript></Span>
            </id>
            <parens xml:lang="ru">
              <apply xml:lang="ru">
                <minus xml:lang="ru" />
                <apply xml:lang="ru">
                  <mult xml:lang="ru" />
                  <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                    <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                  </id>
                  <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                    <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
                  </id>
                </apply>
                <apply xml:lang="ru">
                  <mult xml:lang="ru" />
                  <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                    <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span>
                  </id>
                  <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                    <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                  </id>
                </apply>
              </apply>
            </parens>
          </apply>
        </apply>
        <unitOverride xml:lang="ru">
          <placeholder xml:lang="ru" />
        </unitOverride>
      </eval>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
</region>

8. Используйте функцию norm, чтобы найти норму вектора, содержащего элементы столбца 0 матрицы M.

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<region id="ID0EOALMB" actualWidth="109.26666666666668" actualHeight="69.37360000000001" top="1497.6000000000001" left="38.400000000000006" xml:lang="ru" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="78" xml:lang="ru">
    <define xml:lang="ru" xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">v</id>
      <eval xml:lang="ru">
        <matrix rows="3" cols="1" xml:lang="ru">
          <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
            <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
          </id>
          <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
            <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
          </id>
          <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
            <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
          </id>
        </matrix>
        <unitOverride xml:lang="ru">
          <placeholder xml:lang="ru" />
        </unitOverride>
      </eval>
    </define>
  </math>
</region>

Нажать для копирования этого выражения

Либо вручную вычислите норму вектора v.

Нажать для копирования этого выражения

<region id="ID0E4BLMB" actualWidth="207.04933364629747" actualHeight="36.780453040075308" top="1632.0000000000002" left="38.400000000000006" xml:lang="ru" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="82" xml:lang="ru">
    <define xml:lang="ru" xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve" xml:lang="ru">N</id>
      <eval xml:lang="ru">
        <apply xml:lang="ru">
          <nthRoot xml:lang="ru" />
          <placeholder xml:lang="ru" />
          <apply xml:lang="ru">
            <plus xml:lang="ru" />
            <apply xml:lang="ru">
              <plus xml:lang="ru" />
              <apply xml:lang="ru">
                <pow xml:lang="ru" />
                <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                  <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">a<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                </id>
                <real xml:lang="ru">2</real>
              </apply>
              <apply xml:lang="ru">
                <pow xml:lang="ru" />
                <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                  <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">b<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
                </id>
                <real xml:lang="ru">2</real>
              </apply>
            </apply>
            <apply xml:lang="ru">
              <pow xml:lang="ru" />
              <id labels="VARIABLE" xml:space="preserve" xml:lang="ru" label-is-contextual="true">
                <Span xml:lang="ru" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">c<Subscript xml:lang="ru" xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span>
              </id>
              <real xml:lang="ru">2</real>
            </apply>
          </apply>
        </apply>
        <unitOverride xml:lang="ru">
          <placeholder xml:lang="ru" />
        </unitOverride>
      </eval>
    </define>
    <resultFormat xml:lang="ru">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ru" />
    </resultFormat>
  </math>
</region>