Re: eigenvectors

[John W. Eaton]

On  9-Jun-1999, address@hidden <address@hidden> wrote:

| Ahh, I'd forgotten about the eig function.  I was looking in the help
| under matrix factorizations and eig wasn't listed.  It's under basic
| matrix functions.

Oops.  I think I should move it.

| Thanks,
| 
| On Wed, 9 Jun 1999, Nimrod Mesika wrote:
| 
| > address@hidden wrote:
| > > 
| > >         Q = inv(X)*D*X
| > > 
| > use   [X,D] = eig(Q);
| > 
| > D is a diagonal matrix (the elements are the eigenvalues of Q: lambda1,
| > lambda2, etc..).
| > X is a matrix of eigenvectors.
| > 
| > Actually, since octave returns X as a unitary matrix (a matrix for which
| > inv(A)=A') you also have the simpler expression:
| > 
| > Q = X' * D * X;

Except that I think the expression should be

  Q = X * D * X'

For example, a quick check shows

  octave:15> q = hilb (3);
  octave:16> [x, d] = eig (q)
  x =

    -0.12766   0.54745   0.82704
     0.71375  -0.52829   0.45986
    -0.68867  -0.64901   0.32330

  d =

    0.00269  0.00000  0.00000
    0.00000  0.12233  0.00000
    0.00000  0.00000  1.40832

  octave:17> x*d*x' - q
  ans =

      0.0000e+00    0.0000e+00    5.5511e-17
      0.0000e+00    0.0000e+00    5.5511e-17
      5.5511e-17    5.5511e-17   -5.5511e-17

  octave:18> x'*d*x - q
  ans =

    -0.269717   0.083138  -0.607023
     0.083138   0.294810  -0.573999
    -0.607023  -0.573999  -0.025093


Also note that this diagonalization can fail if you have repeated
eigenvalues.  For example, try q = [1, 2, 3; 0, 1, 2; 0, 0, 3].

jwe



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