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I need a way to compute the inverse of the sum of three matrices:

$(A + BB^T + \beta I)^{-1} $ where $I$ is identity and $\beta$ is a constant.

I am not very familiar with linear algebra, but a quick google search didn't help.

Mike Pierce
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Xavier Hubbard Anderson
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  • Are these specific matrices, or just some general matrices? – Mike Pierce Jul 10 '15 at 02:20
  • If you tell us why you want to find that, we could probably better help. – Spenser Jul 10 '15 at 02:24
  • I don't think there is any reason to suppose that the inverse exists... – A. Thomas Yerger Jul 10 '15 at 02:30
  • Let $C=BB^T+\beta I$ (which is symmetric; not sure if that helps). By [this][1] we have that

    \begin{align} (A+C)^{-1}&= A^{-1}-(I+A^{-1}C)^{-1}A^{-1}CA^{-1} \end{align}

    as long as $A$ is invertible. Not sure if that helps. [1]: http://math.stackexchange.com/questions/17776/inverse-of-the-sum-of-matrices

     – Rocket Man Jul 10 '15 at 02:39
  • I am doing a recursive least squares fit by hand, but i made some alterations to the original derivation, arrived at this step and couldnt continue.

    Let's assume the inverse exist. It should (i think).

    Idk if that helps... I might beable to just make python do this for me without using a closed form. But, i dont think so.

     – Xavier Hubbard Anderson Jul 10 '15 at 04:22
  • This question is hard to tackle without any additional information regarding B or A. One thing you can do is to let C = A + bI and use the Woodbury matrix identity, but this won't get you anywhere at least B or A have some structure. https://en.wikipedia.org/wiki/Woodbury_matrix_identity – Car Loz Jun 22 '20 at 16:56

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