Finding inverse of implicitly defined symmetric matrix

Question

I have an$\ n$ by$\ n$ matrix$\ A$ where $$\ A_{i,j}=\frac{\varepsilon^{i+j+1}}{{i+j+1}} $$ for a general$\ \varepsilon \in \mathbb{R}$. The matrix is symmetric but finding eigenvectors seems to be a difficult path to take (i.e. finding eigendecomposition or SVD) and I'm trying to find a general$\ A^{-1} $. Any suggestions?

Answer 1

$A=\varepsilon DCD$, where $D=\operatorname{diag}(\varepsilon,\varepsilon^2,\ldots,\varepsilon^n)$ and $C$ is the Cauchy matrix given by $c_{ij}=1/(i+j+1)$. So, you may use the formula for the inverse of Cauchy matrix to obtain $A^{-1}$.