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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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    It's a matrix identity trick + triangle ineq. + submultiplicative norm: $(1-F)^{-1}=I+(1-F)^{-1}F$2011-10-25

1 Answers 1

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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\|} } $