[R] A question about positive definite matrix

[Rolf Turner]

On 20/10/2008, at 5:26 PM, rlearner309 wrote:

>
> I know, this is a forum about R.  But I am so desperate of this  
> problem (BTW,
> anyone knows any good Statistics/Math forum to post question like  
> this?):
>
> A and B are both n x n positive definite matrix.
> Denote A > B, if A - B is positive definite.
> I know this is true: if A > B, then A^{-1} < B^{-1}.  But how to  
> prove this?
> I tried to diagonalize A and B, but since they can have different  
> eigen
> structure,... I am stuck here.
> Thanks a lot for any help here.



Your problem intrigued me; I tried to prove the result; couldn't;  
asked a clever
mate; he couldn't; he asked some clever mates, who finally came up  
with the proof
that I have attached in pdf form.

(Anyone who's interested: If the attachment gets removed by the list,  
send me email
and I'll send the pdf file to you directly.)

	cheers,

		Rolf Turner

P. S.  ``WLU'' (in the pdf file) means Wilfrid Laurier University
(Waterloo, Ontario, Canada).



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