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Finding Eigenvectors

Finding Eigenvectors. Some Examples. General Information. Eigenvalues are used to find eigenvectors. The sum of the eigenvalues is called the trace. The product of the eigenvalues is the determinant of the matrix.

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Finding Eigenvectors

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  1. Finding Eigenvectors Some Examples

  2. General Information • Eigenvalues are used to find eigenvectors. • The sum of the eigenvalues is called the trace. • The product of the eigenvalues is the determinant of the matrix. • An EIGENVECTOR of an n x n matrix A is a vector such that , where v is the eigenvector.

  3. Eigenvectors • An eigenvector is a direction for a matrix. • What is important about an eigenvector is its direction. • Every square matrix has at least one eigenvector. • An n x n matrix should have n linearly independent eigenvectors.

  4. Homogeneous Linear SystemsDistinct Eigenvalues

  5. Distinct Eigenvalues (cont) For so that x=0 (row 3), z=0 (row 1), y can be anything. If y=1, one eigenvector is The generalized eigenvector may be written

  6. Distinct Eigenvalues (cont) For so that y=0 (row 3), -x+z=0. If x = 1, one eigenvector is The generalized eigenvector may be written

  7. Solution-Distinct Eigenvalues • The general solution may be written OR

  8. Complex Eigenvalues For

  9. Complex Eigenvalues (cont) • Now for • Multiplying row 1 by 2i and adding to row three gives • Solving, we get z = ( 2i )x and y=0. If x = i, the eigenvector is

  10. Complex Eigenvalues (cont) • The eigenvector was written with a real part and a complex part: • Let be the real part and be the imaginary part. • The eigenvectors corresponding to the complex conjugate pair of eigenvalues may be written:

  11. Complex Eigenvalues (cont) • Don’t forget Euler’s Formula: • For the eigenvectors are

  12. Complex Eigenvalues (cont) • The solution for the given system is

  13. Solving the characteristic equation, we find that Note that there is a repeated eigenvalue. Repeated Eigenvalues

  14. Repeated Eigenvalues (cont) • For , one eigenvector is • For we are able to find two linearly independent eigenvectors: • as eigenvectors. • Solution:

  15. Repeated Eigenvalues (cont) • What happens if it is not possible to find two linearly independent eigenvectors when there is a repeated eigenvalue? • As an example, consider an eigenvalue of multiplicity two with only one eigenvalue associated with this value. A second solution of the form may be found with K and P the required eigenvectors.

  16. Repeated Eigenvectors (cont) • K must be an eigenvector of the matrix A associated with the eigenvalue. To find the second solution, we need to solve • This process may be extended if necessary. For example, if an eigenvalue has multiplicity three and one eigenvalue, K, has been found, then solve and

  17. An Example of Repeated Eigenvalues

  18. An Example of Repeated Eigenvalues • For • It is not possible to find two linearly independent eigenvectors associated with • Solve where P is a column vector.

  19. An Example of Repeated Eigenvalues • With • P= • The solution is • NOTE THE FORM OF THE SOLUTION.

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