About the inverse matrix of the form $(I+cH^{-1})^{-1}$.

Question

Given $(I+cH^{-1})^{-1}$, where $c$ is a constant and $H$ is a $\mathbb{R}^{n\times n}$ matrix. Suppose $(I+cH^{-1})^{-1}$ has a inverse matrix.

Is there any way to calculate $(I+cH^{-1})^{-1}$ only by $H$.

Thank you in advance!

Answer 1

Yes, but that's probably not the way you want.

Suppose $c\ne0$ and $p(x)$ is the characteristic polynomial of $\frac Hc$. Let $p(x-1)=xq(x)+r$ for some polynomial $q$ and some scalar $r$. By Cayley-Hamilton theorem, $(I+\frac Hc)\,q(I+\frac Hc)+rI=0$. Hence $$(I+cH^{-1})^{-1} = \frac Hc\left(I+\frac Hc\right)^{-1}=-\frac H{cr}\,q\left(I+\frac Hc\right).$$