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do we have closed form for $(P+D)^{-1}$, where $P$ is a symmetric positive definite matrix and $D$ is diagonal with positive diagonal elements?

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  • $\begingroup$ I don't know if this meets your needs. $\endgroup$ – thanasissdr Sep 23 '15 at 23:41
  • $\begingroup$ unfortunately no :( $\endgroup$ – Alireza Sep 23 '15 at 23:48
  • $\begingroup$ What exactly are you looking for? In what terms do you want to express the matrix $(P+D)^{-1}$? $\endgroup$ – thanasissdr Sep 23 '15 at 23:53
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    $\begingroup$ the diagonal elements of $D$ are my optimization variables and $P$ is fixed and known, my objective functions involves this inverse, so i need something that does not require getting non defined inverses, something expressed in terms of $D^{-1}$ and $P^{-1}$ (known) , $\endgroup$ – Alireza Sep 23 '15 at 23:57
  • $\begingroup$ I don't think there is any closed-form formula. $(P+D)^{-1}$ is a polynomial in $P+D$, hence also a polynomial in $P^{-1}$ and $D^{-1}$. But the coefficients of this polynomial depend on the eigenvalues of $P+D$, and I don't think you can obtain that information by considering $P$ and $D$ separately. $\endgroup$ – user1551 Sep 24 '15 at 2:27

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