Inverse complex matrix

matrices

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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.

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Author by

yemino

Updated on August 01, 2022

Comments

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