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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?

Alireza
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  • I don't know if this meets your needs. – thanasissdr Sep 23 '15 at 23:41
  • unfortunately no :( – Alireza Sep 23 '15 at 23:48
  • What exactly are you looking for? In what terms do you want to express the matrix $(P+D)^{-1}$? – thanasissdr Sep 23 '15 at 23:53
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    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) , – Alireza Sep 23 '15 at 23:57
  • 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. – user1551 Sep 24 '15 at 02:27
  • @user1551 I thought there must be something neat about this. Because it seems very structured to me. $D$ is diagonal with positive elements and $P$ is positive definite. Can we have a relation ship between eigenvalues of $P+D$ and eigenvalues of $P$ and $D$? – Alireza Sep 24 '15 at 15:10
  • @Alireza You have Weyl's inequality, which give some bounds to the eigenvalues of $P+D$, but you can't infer the precise values of these eigenvalues from the spectra of $P$ and $D$. – user1551 Sep 24 '15 at 19:05
  • @user1551 tnx, but Weyl's inequality is just a bound I wanted sth that includes all eigenvalues of matrices not just min/max – Alireza Sep 25 '15 at 14:42
  • Related : https://math.stackexchange.com/q/2977195/2987 – Rajesh D Oct 30 '18 at 11:22

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