Let's assume I have the matrices $I\in \Re^{nxn}$ which is the identity matrix and moreover both $H,B\in \Re^{nxn}$. I want to find an alternative way to write the following espression: $$(I+HB)^{-1}$$ Exactly I don't want to have any term in which i need to invert something where $B$ appears. For example i tried to use the Matrix Inversion Lemma obtaining: $$(I+HB)^{-1}=I-IH(I+BIH)^{-1}BI=I-H(I+BH)^{-1}B$$ But as you can see in the middle term $(I+BH)^{-1}$ there is an inversion where $B$ appears. Any suggestions?
2026-03-25 12:52:37.1774443157
Inverse without compute inverse
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You can write it using the geometric series, at least formally: $$(I+BH)^{-1}=\sum_{j=0}^{\infty}(-HB)^j,$$ but you'll definitely have some convergence issues to work out. Here, also, as a convention, $(-HB)^0:=I$.