Inverse complex matrix
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.
yemino
Updated on August 01, 2022Comments

yemino over 1 year
I calculated the inverse of an complex matrix $C=A+iB$, where $A,B$ are real matrices and $i^2=1$:
$C^{1}=(A+BA^{1}B)^{1}iA^{1}B(A+BA^{1}B)^{1}$
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?
Thanks!

Ben Grossmann about 10 yearsSo, 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?

yemino about 10 yearsfind some as "$C^{1}$ is invertible if and only if $A$ and $B$ are invertible also", would be ideal.
