# Moore-Penrose inverse invariance

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
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
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?