Пример: Обнаружение левых и правых собственных векторов

Функции > Векторы и матрицы > Характеристики массива > Пример: Обнаружение левых и правых собственных векторов

Во избежание логических несоответствий при выполнении логических сравнений включите Приблизительное равенство (Approximate Equality) в выпадающем списке Параметры расчета (Calculation Options).

1. Определите квадратную действительную матрицу.

2. Используйте основное определение eigenval, чтобы найти значения лямбда матрицы A.

<region id="ID0EMLAFB" actualWidth="220.30666666666667" actualHeight="39.2" align-x="0" align-y="-30.800000000000004" top="345.6" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="28">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <absval />
        <apply>
          <minus />
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id>
          <apply>
            <mult />
            <id xml:space="preserve">λ</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">I</id>
          </apply>
        </apply>
      </apply>
      <command>
        <id xml:space="preserve">factor</id>
      </command>
      <symResult>
        <apply>
          <mult />
          <parens>
            <apply>
              <plus />
              <id labels="*" xml:space="preserve">λ</id>
              <real>3</real>
            </apply>
          </parens>
          <parens>
            <apply>
              <minus />
              <id labels="*" xml:space="preserve">λ</id>
              <real>4</real>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <formattingOverride BackgroundColor="#00FFFFFF" />
  </math>
</region>

3. Либо используйте функцию eigenvals, чтобы найти значения лямбда матрицы A.

Поиск правых собственных векторов

1. Используйте функцию eigenvecs, чтобы найти правые (по умолчанию) собственные векторы матрицы A.

2. Покажите, что отношение A x Rn = λn x Rn истинно.

<region id="ID0EDVAFB" actualWidth="120.21666666666667" actualHeight="44.2" align-x="0" align-y="-25.51" top="940.80000000000018" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="35"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0ETVAFB" actualWidth="123.25" actualHeight="44.2" align-x="0" align-y="-25.51" top="940.80000000000018" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="36"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0EEWAFB" actualWidth="131.55333333333334" actualHeight="27.791200000000003" align-x="0" align-y="-17.200000000000003" top="940.80000000000018" left="384.00000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="37"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> </resultFormat> </math> </region>
<region id="ID0EXWAFB" actualWidth="130.96666666666664" actualHeight="44.2" align-x="0" align-y="-25.51" top="998.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EGXAFB" actualWidth="134" actualHeight="44.2" align-x="0" align-y="-25.51" top="998.40000000000009" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="39"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EWXAFB" actualWidth="131.55333333333334" actualHeight="27.791200000000003" align-x="0" align-y="-17.200000000000003" top="998.40000000000009" left="384.00000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="40"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">R<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>

Отношения истинны.

Поиск левых собственных векторов

1. Используйте функцию eigenvecs, чтобы найти левые собственные векторы матрицы A.

2. Покажите, что отношение LnT x A = λn x LnT истинно.

<region id="ID0ER5AFB" actualWidth="169.52333333333331" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1401.6000000000001" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0EB6AFB" actualWidth="184.89" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1401.6000000000001" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="45"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> <apply> <transpose /> <parens> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </parens> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0ES6AFB" actualWidth="146.19333333333336" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1401.6000000000001" left="441.60000000000008" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="46"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript></Span> </id> </apply> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>
<region id="ID0EFABFB" actualWidth="180.27333333333331" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1459.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="47"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EUABFB" actualWidth="195.64" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1459.2000000000003" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="48"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> <apply> <transpose /> <parens> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </parens> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EEBBFB" actualWidth="146.19333333333336" actualHeight="29.943200000000004" align-x="0" align-y="-19.352" top="1459.2000000000003" left="441.60000000000008" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="49"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">λ<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">L<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript></Span> </id> </apply> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>

Отношение истинно.

Было ли это полезно?