# Moore-Penrose pseudo inverse

I have a matrix Z n times p with p>n

I have A a diagonal matrix with positive entries

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

the MP inverse of Z^T Z

and the MP inverse of A Z^T Z A

what i am looking for is the follwing: suppose I knwo the MP of $Z^T Z$ and I know $A$, can I get as a function of those two things the MP inverse of $A Z^T Z A$?

-
could you clarify your question – optima Dec 30 '10 at 17:48
Please read the FAQ! This is not really at the level suited for MO. You might have more luck at: math.stackexchange.com – Suvrit Dec 30 '10 at 18:06
For the people who think this is trivial: dx.doi.org/10.1137/1006007 – J. M. Dec 31 '10 at 5:15
Ok, it seems I was a bit hasty in dismissing this question. It is not as trivial as I thought! – Suvrit Dec 31 '10 at 15:04

## 2 Answers

"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"

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

-

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.

-
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 – Liliana Dec 30 '10 at 21:04