I don't understand the following implication:
$$\|(I-E)^{-1}\| ≤ 1 + \|E\|\|(I-E)^{-1}\| \implies \|(I-E)^{-1}\| ≤ \dfrac{1}{1-\|E\|}$$
Understanding an implication involving operator norms:
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linear-algebra
matrices
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0Where did you encounter this inequality? Is your problem with finding the inequality in the first place, or is the problem with the implication? – 2017-01-28
1 Answers
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Rearrange the inequality as follows: $$ \|(I-E)^{-1}\| ≤ 1 + \|E\|\|(I-E)^{-1}\| \implies\\ \|(I-E)^{-1}\| - \|E\|\|(I-E)^{-1}\| \leq 1 \implies\\ (1 - \|E\|)\;\|(I-E)^{-1}\| \leq 1 \implies\\ \|(I-E)^{-1}\| \leq \frac 1{1 - \|E\|} $$