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Finding Eigenvectors - PowerPoint PPT Presentation


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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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general information
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.
eigenvectors
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.
distinct eigenvalues cont
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

distinct eigenvalues cont6
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

solution distinct eigenvalues
Solution-Distinct Eigenvalues
  • The general solution may be written

OR

complex eigenvalues cont
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
complex eigenvalues cont10
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:
complex eigenvalues cont11
Complex Eigenvalues (cont)
  • Don’t forget Euler’s Formula:
  • For the eigenvectors are
complex eigenvalues cont12
Complex Eigenvalues (cont)
  • The solution for the given system is
repeated eigenvalues cont
Repeated Eigenvalues (cont)
  • For , one eigenvector is
  • For we are able to find two linearly independent eigenvectors:
  • as eigenvectors.
  • Solution:
repeated eigenvalues cont15
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.
repeated eigenvectors cont
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
an example of repeated eigenvalues18
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.
an example of repeated eigenvalues19
An Example of Repeated Eigenvalues
  • With
  • P=
  • The solution is
  • NOTE THE FORM OF THE SOLUTION.