Example: Finding the Left and Right Eigenvectors

Functions > Vector and Matrix > Array Characteristics > Example: Finding the Left and Right Eigenvectors

To avoid logical mismatches when performing boolean comparisons, enable Approximate Equality in the Calculation Options drop-down list.

1. Define a square real matrix.

2. Use the basic definition of eigenval to find the lambdas of matrix A.

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  <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. Alternatively, use function eigenvals to find the lambdas of matrix A.

Finding the Right eigenvectors

1. Use function eigenvecs to find the right (default) eigenvectors of matrix A.

2. Show that the relationship A x Rn = λn x Rn is true.

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The relationships are true.

Finding the Left eigenvectors

1. Use function eigenvecs to find the left eigenvectors of matrix A.

2. Show that the relationship LnT x A = λn x LnT is true.

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The relationship is true.