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Topic: Does this matrix function have real eigenvalues?
Replies: 9   Last Post: Aug 1, 2012 3:43 AM

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karl

Posts: 398
Registered: 8/11/06
Re: Does this matrix function have real eigenvalues?
Posted: Jul 28, 2012 2:42 AM
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Am 28.07.2012 07:25, schrieb Paul:
> On Jul 27, 11:24 pm, karl <oud...@nononet.com> wrote:
>> Am 28.07.2012 04:21, schrieb Paul:
>>

>>> Let G = I - inv(W) * B where W and B are real symmetric positive
>>> definite matrices.
>>> In simulations it appears that G always has real eigenvalues, though G
>>> is not necessarily symmetric or positive definite.

>>
>>> I wonder if it can be proved in general that G has real eigenvalues?
>>> Bonus question: Is there a simple relationship between the eigenvalues
>>> of G and those of W and B?

>>
>>> Best,
>>> Paul

>>
>> G= inv(W)*(W-B)= inv(W)*W-inv(W)*B=I-inv(W)*B.
>> Inv(W) and W-B are symmetric, therefore their product will be symmetric too AFAIS.
>>
>> Karl

>
> Thanks. Your re-expression of G is correct. However, the product of
> symmetric matrices is not necessarily symmetric.
>


Ok, too fast. Look here:


http://www.physicsforums.com/showthread.php?t=6676

This condition is fulfilled, since two positive definit symmetric matrices can be jointly diagonalized:


http://en.wikipedia.org/wiki/Positive-definite_matrix#Simultaneous_diagonalization

You also need: If two symmetric matrices commute, their product is symmetric.

Karl



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