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Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then

$$\lVert (I-A)^{-1}\rVert \leq \frac{\lVert I\rVert-(\lVert I\rVert-1)\lVert A\rVert}{1-\lVert A\rVert}.$$

This is the second part of the problem. I finally figured out the first part, but I am having trouble starting this one. Does anyone know what inequality I should start with to get this? Or the procedure I should take? I think with a little help I should be able to figure this one out.

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What is $B$, all matrices of the same size? Could you also maybe tell us part 1 and how you solved it? – bgins Apr 12 '12 at 16:04
Are you working with any matrix norm, or a particular one? – bgins Apr 12 '12 at 16:13
First show that $\|(I-A)^{-1}\|\leq \frac{1}{1-\|A\|}$ – Kuku Apr 12 '12 at 16:55
I'm working with just the standard norm, so it could be any norm. – Sarah Martin Apr 12 '12 at 19:52
up vote 0 down vote accepted

I am dealing with the standard norm, so the actual norm isn't defined. I believe I solved it though. Correct me if I'm wrong, but expanding (I−A)−1=I+A−A2+...+/−(Ap−1) for the pth partial sum. Then ||(I−A)−1|| ≤||A||+||A2||+...+||Ap−1|| ≤||A||+||A||2+...+||A||p−1 =[1−||A||p]/(1−||A||) ≤1/(1−||A||) =(1+||A||−||A||)/(1−||A||) =[||I||−(||I||−1)||A||]/(1−||A||)

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