From help-octave-request at bevo dot che dot wisc dot edu  Wed Jun  9 16:17:31 1999
Subject: Re: eigenvectors
From: Thomas Shores <tshores at mathstat dot unl dot edu>
To: heberf at calvin dot wustl dot edu
Cc: help-octave at bevo dot che dot wisc dot edu
Date: Wed,  9 Jun 1999 16:15:55 -0500

Whoops,  the story is a bit more complicated than that.  Herber Farnsworth 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.
>
>Thanks,
>
>On Wed, 9 Jun 1999, Nimrod Mesika wrote:
>
>> heberf at calvin dot wustl dot edu 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;
> >
>> -- Nimrod.
> >


Actually, it's a bit more complicated than that.  Not every matrix is even  
diagonalizable, let alone unitarily diagonalizable.  Any *real symmetric*  
matrix, such as a Hilbert matrix, is automatically unitarily (X^{-1}=X')  
diagonalizable, whether it has repeated eigenvalues or not. On the other  
hand,  a matrix like a = [1,2;0,3] is diagonalizable, but not unitarily  
diagonalizable.  Worse yet, a matrix like a = [1,1;0,1] is simply not  
diagonalizable at all.  In the first case, octave will return a unitary  
matrix.  In the second case, it will return a matrix whose columns are unit  
length eigenvectors.  And in the third case it will return a completely  
incorrect matrix (none can work) with unit length columns.  BTW, if you do  
want to review these linear algebra concepts, I keep a copy of a text I'm  
writing on the web in my linear algebra home page  
(http://www.math.unl.edu/~tshores/linalgtext.html), so feel free to use it  
for a quick reference.

Tom Shores




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