Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I calculated the inverse of an complex matrix $C=A+iB$, where $A,B$ are real matrices and $i^2=-1$:


my question is: what assumptions must be met $A$ and $B$ to have this inverse?

Obviously, must be $A^{-1}$ and $(A+BA^{-1}B)^{-1}$, but there exist a way to characterize this?


share|cite|improve this question
So, it seems your question is as follows: "given an invertible matrix $A$, for which matrices $B$ is $A + BA^{-1}B$ invertible?" What kind of conditions on $B$ are you looking for? Something about the rank of $B$, perhaps? Maybe a condition on the eigenvalues/eigenvectors? Do you have anything in mind that would be particularly helpful? – Omnomnomnom Nov 3 '13 at 22:11
find some as "$C^{-1}$ is invertible if and only if $A$ and $B$ are invertible also", would be ideal. – yemino Nov 3 '13 at 22:37

I suppose you are asking for a reformulation of the condition that both $A$ and $A+BA^{-1}B$ are invertible. Note that this is different from asking when will $C$ be invertible, because $C$ can be invertible when $A$ is singular (e.g. consider $C=iI$).

Since $A+BA^{-1}B=A\left(I+(A^{-1}B)^2\right)$, you may rewrite the condition that both $A$ and $A+BA^{-1}B$ are invertible as

$A$ is invertible and $\pm i$ is not an eigenvalue of $A^{-1}B$,

but I am not sure if this is really useful. Alternatively, viewing $A+BA^{-1}B$ as a Schur complement, one can obtain another equivalent condition:

both $A$ and $\pmatrix{A&-B\\ B&A}$ are invertible.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.