Inverse of a sum of positive definite matrices

Question

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ (A+B)^{-1} = L(I+L^TBL)^{-1}L^T \\ |A+B| = |A| \, |I+L^T B L| $$ Are they correct? And if so, how do you show they're true? I suppose they can be derived from the Woodbury formula and the matrix determinant lemma, but I tried and I couldn't prove it...

Both identities follow from $A+B = L^{-T} L^{-1}+B = L^{-T} ( I + L^T B L ) L^{-1}$. — copper.hat, May 21 '14 at 18:23
Thanks. Your comment was the most useful. I would've accepted it as an answer! — , May 21 '14 at 18:36

score 1 · Answer 1 · answered May 21 '14 at 18:17

1

Hint: verify $(A+B)^{-1}(A+B)=I$, $|X|=1/|X^{-1}|$.

answered May 21 '14 at 18:17

Easy

4,485

score 1 · Accepted Answer · answered May 21 '14 at 18:20

1

Hint: notice that: $$L^{-1}(A+B)^{-1}L^{-T}=(I+L^TBL)^{-1}$$ Also $$L^{-1}(A+B)^{-1}L^{-T}=(L^T(A+B)L)^{-1}$$

answered May 21 '14 at 18:20

enigne

813

Inverse of a sum of positive definite matrices

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