Further Examples: Eigensystem

These are the eigenvalues of an exact matrix m1.

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These are the eigenvectors of m1.

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This lists the eigenvalues and the eigenvectors for m1. The eigenvalues are listed first.

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This verifies the result.

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Here are two inexact matrices.

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This gives the generalized eigenvalues and eigenvectors of m2 with respect to m3.

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This checks the result.

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Here are two more matrices.

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One of the generalized eigenvalues is .

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This checks the values of the two finite eigenvalues.

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The reciprocals of the eigenvalues of are equal to the eigenvalues of .

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Using Eigensystem to diagonalize a matrix

We can use Eigensystem to diagonalize a matrix that has distinct eigenvalues.

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The eigenvalues give the diagonalized form.

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The columns of the change of basis matrix ch are the eigenvectors.

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We check the diagonalization.

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Using Eigenvalues to find the roots of a polynomial

Solve uses the Jenkins-Traub algorithm to find approximate numeric roots to a polynomial in one variable. Another method uses the eigenvalues of the companion matrix for that polynomial.

This gives the companion matrix of a polynomial.

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Here is a polynomial and its companion matrix.

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This sorts the eigenvalues of the polynomial's companion matrix.

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Here is a comparison with the Mathematica default technique of finding the roots of a polynomial.

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When a matrix is defective

The matrix m1 is defective since the number of linearly independent eigenvectors is less than .

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This lists the eigenvalues and the eigenvectors for m1. The eigenvalues are listed first.

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Although the vectors in vectors are the eigenvectors of the matrix m1, they provide only one linearly independent vector, . The other computed vectors are linearly dependent on . It follows that m1 is not invertible.

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Note that the computations are inexact, and the eigenvectors are also inexact. Here it is evident that the terms of order are roundoff errors. However, it is not always correct to use Chop since Chop may remove terms which are not roundoff errors.

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