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I have been given the following question as part of an assignment: "Consider the $N \times N$ matrices $H_0$ and $V$; consider the matrix $G$ defined by $G=(zI - H_0 - V)^{-1}$ where $z$ is a scalar and $I$ is the identity matrix. Let $G_0 = (zI - H_0)^{-1}$. Show that $G$ can be written as

$$ G = G_0 \sum_{n=0}^{\infty} (V G_0)^n = \sum_{n=0}^{\infty} (G_0 V)^n G_0 $$

It also says to formally use the following geometric series on the $ G=(zI - H_0 - V)^{-1} $ equation:

$$ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n, \quad \mbox{for } |x| < 1 $$

I am unsure how exactly to directly apply this geometric series onto the $ G=(zI - H_0 - V)^{-1}$ equation. I suspect that there is some property to relate $VG_0$ and $G$? I would appreciate if someone would be able to steer me in the right direction. Thanks.

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Hint: The geometric series in matrix form is: $$ (I-A)^{-1}=\sum_{n=0}^\infty A^n $$ when $\|A\|<1$. This can also help.

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    I am unsure how exactly to now apply that series. Would I set A = G or G0? Or VG0? – 2017-02-02