Matrix Norm Identity Derivation

I am having trouble figuring out where something in a book I am reading is coming from. (The book is Matrix Computations by Golub and Van Loan, 3rd edition, p.58.) It will probably be obvious once someone points it out to me, but until then I am stuck. It stems from the following lemma (which I understand and can follow):

If $F \in \mathbb{R}^{n \times n}$ and $\|F\|_p < 1$, then $I-F\,\,$ is nonsingular and

$$\left( I-F \right )^{-1} = \sum_{k=0}^{\infty}{F^k}$$

with

$$\|\left( I-F \right )^{-1} \|_p \leq { {1} \over {1-\|F\|_p} }.$$

As a consequence of the above,

$$\| \left( I - F \right )^{-1} - I \|_p \leq { {\|F\|_p} \over {1-\|F\|_p}}.$$

It's the last inequality that I cannot figure out how to derive.

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Which book? ${}$ – J. M. Oct 25 '11 at 17:40
Golub and Van Loan, Matrix Computations, 3rd edition, p. 58. – anonstudent45678 Oct 25 '11 at 17:46
It's a matrix identity trick + triangle ineq. + submultiplicative norm: $(1-F)^{-1}=I+(1-F)^{-1}F$ – user13838 Oct 25 '11 at 18:26

1 Answer

Thanks to percusse in the comments above. The identity provided makes it rather simple:

$$\left( I - F \right)^{-1} - I = \left( I - F \right)^{-1} F$$

so

$$\|(I-F)^{-1} - I\| \leq \|F\| \cdot \|(I-f)^{-1}\| \leq { {\|F\|} \over {1-\|F\|} }$$

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