例:左と右の固有ベクトルを求める

関数 > ベクトルと行列 > 配列の特徴 > 例:左と右の固有ベクトルを求める

例:左と右の固有ベクトルを求める

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1. 正方実数行列を定義します。

2. eigenval の基本定義を使用して、行列 A のラムダを求めます。

<region id="ID0EPJPFB" actualWidth="220.30666666666667" actualHeight="39.2" top="211.20000000000002" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="26">
    <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>
              <minus />
              <id labels="VARIABLE" xml:space="preserve">λ</id>
              <real>4</real>
            </apply>
          </parens>
          <parens>
            <apply>
              <plus />
              <id labels="VARIABLE" xml:space="preserve">λ</id>
              <real>3</real>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <formattingOverride BackgroundColor="#00FFFFFF" />
  </math>
</region>

3. または、関数eigenvalsを使用して、行列 A のラムダを求めます。

右の固有ベクトルを求める

1. 関数eigenvecsを使用して行列 A の右 (デフォルト) の固有ベクトルを求めます。

2. A x Rn と λn x Rn の間の偏差を確認します。

<region id="ID0EJTPFB" actualWidth="120.21666666666667" actualHeight="44.2" top="921.60000000000014" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="33"> <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="ID0EZTPFB" actualWidth="123.25" actualHeight="44.2" top="921.60000000000014" left="172.8" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="34"> <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="ID0EKUPFB" actualWidth="146.17666666666668" actualHeight="44.2" top="921.60000000000014" left="345.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="35"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <minus /> <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="ID0E4UPFB" actualWidth="130.96666666666664" actualHeight="44.2" top="979.20000000000016" left="0" 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">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="ID0EMVPFB" actualWidth="134" actualHeight="44.2" top="979.20000000000016" left="172.8" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="37"> <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="ID0E3VPFB" actualWidth="146.17666666666668" actualHeight="44.2" top="979.20000000000016" left="345.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <minus /> <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>

偏差はゼロです。

左の固有ベクトルを求める

1. 関数 eigenvecs を使用して行列 A の左の固有ベクトルを求めます。

2. LnT x A と λn x LnT の間の偏差を確認します。

<region id="ID0E23PFB" actualWidth="169.52333333333331" actualHeight="29.943200000000004" top="1382.4" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="42"> <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="ID0EL4PFB" actualWidth="184.89" actualHeight="29.3712" top="1382.4" left="172.8" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="43"> <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="ID0E34PFB" actualWidth="174.46333333333337" actualHeight="29.943200000000004" top="1382.4" left="403.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <minus /> <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="ID0EP5PFB" actualWidth="180.27333333333331" actualHeight="29.943200000000004" top="1440.0000000000002" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="45"> <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="ID0E55PFB" actualWidth="195.64" actualHeight="29.3712" top="1440.0000000000002" left="172.8" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="46"> <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="ID0EO6PFB" actualWidth="174.46333333333337" actualHeight="29.943200000000004" top="1440.0000000000002" left="403.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="47"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <minus /> <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>

偏差はゼロです。