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Let $A\in\mathbb{C}^{n\times n}$ be non-singular. I have to prove that the condition number of A with the $p$-norm, $\kappa_p(A)$ ($1\leq p\leq\infty$), satisfies $$\kappa_p(A) \geq \frac{\|A\|_p}{\|A-B\|_p},$$ for every singular matrix $B\in\mathbb{C}^{n\times n}$.

I don't know what to do with this and I'd appreciate any help someone could give me.

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We have $$ \begin{split} \|A-B\| &= \max_{\|x\|=1}\|(A-B)x\| \geq \max_{\|x\|=1\\ Bx=0}\|(A-B)x\| \\ &= \max_{\|x\|=1\\ Bx=0}\|Ax\| \geq \min_{\|x\|=1}\|Ax\| = \frac{1}{\|A^{-1}\|}. \end{split} $$ Invert the inequality and multiply both sides by $\|A\|$. This holds for any operator norm (not just the $p$-norms).

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  • $\begingroup$ Thank you very much, this was very helpful! $\endgroup$ Apr 12, 2018 at 21:49

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