# 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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