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I have an $n\times p$ matrix $Z$ with $p\gt n$.

I have a diagonal matrix $A$ with positive entries.

Is there is a known way to determine the MP inverse of $A Z^T Z A$, if I know $A$ and the MP inverse of $Z^T Z$.

Thanks

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Does $\mathbf Z^T$ have full rank? –  J. M. Dec 31 '10 at 0:53
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This article by R.E. Cline gives an expression for the pseudoinverse of the product of two matrices: $$(\mathbf A\mathbf B)^+=(\mathbf A^+\mathbf A\mathbf B)^+(\mathbf A\mathbf B(\mathbf A^+\mathbf A\mathbf B)^+)^+$$ –  J. M. Dec 31 '10 at 5:12
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Yes, Z has full rank, and A is positive definite, but using the formula I do not get how i can compute $(A Z^T Z A)^+ $ as a function of $(Z^T Z)^+ $ and $A$... or am I doing something wrong?

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If you are the same person as the OP, you might want to ask a moderator to merge your accounts. –  Arturo Magidin Dec 31 '10 at 19:06
    
@arturo how does one go about contacting a moderator on this list? –  ronaf Jan 1 '11 at 2:18
    
@ronaf: for a specific post or comment, With enough reputation, you can flag and call up moderator attention. Doesn't take much (15 pts). There should be an option of "needs moderator attention". Otherwise, you can try posting in meta. –  Arturo Magidin Jan 1 '11 at 2:28
    
@arturo the flag option i can conjure up for a comment will identify it as noise, offensive or spam. i agree that there should be an option "needs moderator attention". but as this doesn't seem to be a list feature at present - i suppose one needs to contact a moderator to suggest adding it. sounds like a nice catch 22 to me. –  ronaf Jan 1 '11 at 2:46
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@ronaf: that's for comments. If you flag a post, there is "needs moderator attention". And there's always meta: there should be a link at the very top that says "meta", just left of "faq". You can post a question there. –  Arturo Magidin Jan 1 '11 at 2:56
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