Inverse of a special block matrix

by Soumitra   Last Updated August 13, 2019 18:20 PM

I have a special $NM \times NM$ matrix of the form

\begin{align*} S = \left[ \begin{array}{cccc} V + \lambda I & V & \cdots & V \\ V & V + \lambda I & \cdots & V \\ \cdots & \cdots & \cdots & \cdots \\ V & V & \cdots & V + \lambda I \end{array} \right] \end{align*} where $V$ is a symmetric $N\times N$ matrix and $I$ is an identity matrix of size $N$.

I want to know if there is any way to express $S^{-1}$ in a simpler form involving $V^{-1}$ and $(V+\lambda I)^{-1}$.



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