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## Re: eigenvectors

 From: Thomas Shores Subject: Re: eigenvectors 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:
>
>> 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;
> >
>> -- 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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