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Suppose we have a matrix $XX^T + D$ and we want to calculate its inverse. $D$ is a diagonal matrix. $X$ is a symmetric matrix.

We know the inverse of $X_1X_1^T, X_2X_2^T, \ldots, X_nX_n^T$, where $X_1X_1^T, X_2X_2^T, \ldots, X_nX_n^T$ are the blocks of the $XX^T$.

Is there any efficient way to calculate $(XX^T + D)^{-1}$?

Thanks a lot.

Siong Thye Goh
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yx131
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  • There is the Sherman Morrison formula for the inverse. Check out its wiki page. – StubbornAtom Aug 08 '17 at 09:51
  • Thanks for your reply. I know the Sherman Morrison formula. However, the problem is that D is a full rank diagonal matrix, and $XX^T$ is also a full rank matrix. – yx131 Aug 08 '17 at 13:35

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