# matrix inverse for sum of positive definite and diagonal matrice

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?

• 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
• 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 2:27