Finding Eigenvectors with Close Eigenvalues

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SUMMARY

The discussion focuses on finding eigenvectors for a matrix with closely spaced eigenvalues derived from the characteristic polynomial -2+x-2x^2-x^3. The eigenvalues identified are -2.659, 0.329-0.802i, and 0.329+0.802i. The method involves substituting each eigenvalue into the matrix equation (A-xI)v=0 and reducing to row echelon form. A common issue arises when the eigenvalues are not sufficiently distinct, leading to an identity matrix, indicating the need for alternative methods or better approximations for eigenvalues.

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  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix operations and row echelon form
  • Knowledge of complex numbers and their properties
  • Experience with characteristic polynomials
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  • Explore numerical methods for eigenvalue approximation
  • Learn about the QR algorithm for finding eigenvalues
  • Investigate the use of perturbation theory in eigenvalue problems
  • Study the implications of eigenvalue sensitivity in matrix analysis
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Students and professionals in mathematics, engineering, and computer science who are working with linear algebra, particularly in applications involving eigenvalues and eigenvectors.

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Homework Statement



Given the characteristic polynomial -2+x-2x^2-x^3, find the eigenvalues and eigenvectors of the matrix [-1, -1, 0] [1, 1, 1] [3, 1, -2]

Homework Equations


The Attempt at a Solution



The eigenvalues are -2.659, 0.329-.802i, and 0.329+.802i. Next you plug each eigenvalue into the matrix A-xI to solve the system (A-xI)v=0 and find the eigenvectors. Then you solve the system by reducing the matrix to row echelon form. However, when I do that I get the identity matrix. So then what are the eigenvectors?
 
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That occurs often when the eigenvalues are not close enough. You either need to switch to a less sensitive method, or find closer approximations to the eigenvalues.
 

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