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We know for $A \in \mathcal M_n$ and every $\epsilon > 0$, there exists a submultiplicative matrix norm $\| \cdot\|_m$ such that $\|A\|_m \le \rho(A) + \varepsilon$. Is this norm vector norm induced? In other words, could we define some norm on $\mathbb C^n$ such that $\|A\|_m = \sup_{\|x\| = 1} \|Ax\|$ for $x \in \mathbb C^n$? Thanks.

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Indeed! Let $S$ be the matrix for which $A=SJS^{-1},$ where $J$ is the Jordan canonical form of $A.$ Let $E=\mathrm{diag}(1,\varepsilon,\varepsilon^{2},\ldots,\varepsilon^{n-1}).$ Then $A=SEJ'E^{-1}S^{-1},$ where $J'$ is like the Jordan canonical form of $A$, but wherever $J$ had a one on its first superdiagonal, $J'$ has a $\varepsilon.$ Let $V=SE.$ Then if we let $\|x\|:=\|V^{-1}x\|_{\infty},$ we see that \begin{align*} \|A\|&=\sup_{\|x\|=1}\|Ax\|\\ &=\sup_{\|V^{-1}x\|_{\infty}=1}\|V^{-1}Ax\|_{\infty}\\ &=\sup_{\|y\|_{\infty}=1}\|V^{-1}AVy\|_{\infty}\\ &=\sup_{\|y\|_{\infty}=1}\|J'y\|_{\infty}=\rho(A)+\varepsilon. \end{align*}

I leave it to you to show that $\|V^{-1}\cdot\|_{\infty}$ defines a valid vector norm on $\mathbb{C}^{n}$, and the last equality.

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  • $\begingroup$ May I ask your reference for this construction? I want to reference it. I am using Matrix Analysis by Horn & Johnson, but it seems like they are not concerned with the vector norm. Thanks. $\endgroup$ – user1101010 Feb 6 '18 at 22:14
  • $\begingroup$ I'm 99% sure all of the supporting material is in H&J, but some of it might be in exercises. I put the material together to make this argument myself, but I'm sure that it's not new. I think citing H&J would be sufficient. $\endgroup$ – RideTheWavelet Feb 7 '18 at 20:15
  • $\begingroup$ For me, the vector norm is more of interest. But in H&J (as least in the main body), it does not been mentioned. $\endgroup$ – user1101010 Feb 7 '18 at 21:46
  • $\begingroup$ Okay, I don't think I was understanding before. In the chapter on norms, there should be an exercise asking you to prove that for a nonsingular matrix $S$ and vector norm $\|\cdot\|$, $\|S^{-1}\cdot\|$ defines a vector norm. $\endgroup$ – RideTheWavelet Feb 8 '18 at 20:40

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