karl
Posts:
213
Registered:
8/11/06
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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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