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.
--
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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). ###################################################################### Attention: This e-mail message is privileged and confidential. If you are not the intended recipient please delete the message and notify the sender. Any views or opinions presented are solely those of the author. This e-mail has been scanned and cleared by MailMarshal www.marshalsoftware.com ###################################################################### -------------- next part -------------- A non-text attachment was scrubbed... Name: pos def.pdf Type: application/pdf Size: 35541 bytes Desc: not available URL: <https://stat.ethz.ch/pipermail/r-help/attachments/20081024/24e9629d/attachment.pdf>
Sender: r-help-bounces at r-project.org On-Behalf-Of: r.turner at auckland.ac.nz Subject: Re: [R] A question about positive definite matrix Message-Id: <9EC7E590-AA42-4CCC-9906-8C47C43CC5AE at auckland.ac.nz> Recipient: ngottlieb at marinercapital.com -------------------------------------------------------- This information is being sent at the recipient's request or with their specific understanding. The recipient acknowledges that by sending this information via electronic means, there is no absolute assurance that the information will be free from third party access, use, or further dissemination. This e-mail contains information that is privileged and/or confidential and may be subject to legal restrictions and penalties regarding its unauthorized disclosure or other use. You are prohibited from copying, distributing or otherwise using this information if you are not the intended recipient. Past performance is not necessarily indicative of future results. This is not an offer of or the solicitation for any security which will be made only by private placement memorandum that may be obtained from the applicable hedge fund. If you have received this e-mail in error, please notify us immediately by return e-mail and delete this e-mail and all attachments from your system. Thank You. -------------- next part -------------- An embedded message was scrubbed... From: Rolf Turner <r.turner at auckland.ac.nz> Subject: Re: [R] A question about positive definite matrix Date: Fri, 24 Oct 2008 08:29:48 +1300 Size: 54614 URL: <https://stat.ethz.ch/pipermail/r-help/attachments/20081023/36415008/attachment-0003.mht>
Sender: r-help-bounces at r-project.org On-Behalf-Of: r.turner at auckland.ac.nz Subject: Re: [R] A question about positive definite matrix Message-Id: <9EC7E590-AA42-4CCC-9906-8C47C43CC5AE at auckland.ac.nz> Recipient: nalbicelli at tricadiacdpcmanagement.com -------------------------------------------------------- This information is being sent at the recipient's request or with their specific understanding. The recipient acknowledges that by sending this information via electronic means, there is no absolute assurance that the information will be free from third party access, use, or further dissemination. This e-mail contains information that is privileged and/or confidential and may be subject to legal restrictions and penalties regarding its unauthorized disclosure or other use. You are prohibited from copying, distributing or otherwise using this information if you are not the intended recipient. Past performance is not necessarily indicative of future results. This is not an offer of or the solicitation for any security which will be made only by private placement memorandum that may be obtained from the applicable hedge fund. If you have received this e-mail in error, please notify us immediately by return e-mail and delete this e-mail and all attachments from your system. Thank You. -------------- next part -------------- An embedded message was scrubbed... From: Rolf Turner <r.turner at auckland.ac.nz> Subject: Re: [R] A question about positive definite matrix Date: Fri, 24 Oct 2008 08:29:48 +1300 Size: 54567 URL: <https://stat.ethz.ch/pipermail/r-help/attachments/20081023/5768880b/attachment-0005.mht>