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Let $C$ be a positive definite matrix, $D$ be a diagonal matrix with all elements being positive and $A=C+D$. By Woodbury matrix identity, we have $A^{-1}=C^{-1}-C^{-1}(D^{-1}+C^{-1})^{-1}C^{-1}$ or $A^{-1}=D^{-1}-D^{-1}(C^{-1}+D^{-1})^{-1}D^{-1}$. However, we have no idea about $(D^{-1}+C^{-1})^{-1}$. My question is as followings:

  1. Is there is closed form of $A^{-1}$ in terms of $C$(or $C^{-1}$) and $D$(or $D^{-1}$)?
  2. If no, is there a explicit relation between $A^{-1}$ and $C$ and $D$ through their eigenvalues and eigenvectors?

Thanks!

Git Gud
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lensincet
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  • What is your concept of "explicit relation"? I dont think there is one. What do you want to achieve? – kjetil b halvorsen Mar 05 '14 at 08:34
  • I'm dealing with an optimization problem whose objective function consisting of $D$ in terms of $A^{-1}$. The optimization variables are the elements in $D$. Here, the "explicit relation" that i expect is that i can deal with the reduced objective without $A^{-1}$ at least, and maybe the the optimization problem can therefore be solved analytically. – lensincet Mar 05 '14 at 08:43
  • Then maybe you should post that problem? – kjetil b halvorsen Mar 05 '14 at 10:37

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