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I have an $n\times p$ matrix $Z$ with $p>n$

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

I would like to know if there is a known relation (as a function of $A$) between

the Moore-Penrose inverse of $Z^T Z$

and the Moore-Penrose inverse of $A Z^T Z A$

what i am looking for is the following: suppose I know the Moore-Penrose of $Z^T Z$ and I know $A$. Can I get, as a function of those two things, the Moore-Penrose inverse of $A Z^T Z A$?

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  • $\begingroup$ could you clarify your question $\endgroup$
    – optima
    Dec 30, 2010 at 17:48
  • $\begingroup$ Please read the FAQ! This is not really at the level suited for MO. You might have more luck at: math.stackexchange.com $\endgroup$
    – Suvrit
    Dec 30, 2010 at 18:06
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    $\begingroup$ For the people who think this is trivial: dx.doi.org/10.1137/1006007 $\endgroup$ Dec 31, 2010 at 5:15
  • $\begingroup$ Ok, it seems I was a bit hasty in dismissing this question. It is not as trivial as I thought! $\endgroup$
    – Suvrit
    Dec 31, 2010 at 15:04

2 Answers 2

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"what i am looking for is what is suppose I know the MP of Z^T Z, how can i get the MP of A Z^T Z A using A and the MP of Z^T Z? thanks"

$(A Z^T Z A)^\dagger = (A ((Z^TZ)^\dagger)^\dagger A)^\dagger$

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If by 'knwon relation' you mean whether you can convert between these matrices unambiguously then the answer is yes. The Moore-Penrose pseudoinverse always exists and is unique, and A is nonsingular. That is all you need.

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    $\begingroup$ what i am looking for is what is suppose I know the MP of Z^T Z, how can i get the MP of A Z^T Z A using A and the MP of Z^T Z? thanks $\endgroup$
    – Liliana
    Dec 30, 2010 at 21:04

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