- #1

mathmari

Gold Member

MHB

- 5,049

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Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ?

I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or $\|G\|_{1}=1$ and $\rho (G)=0.1<1$ in these both cases the iteration method converges, or not? (Wondering)

I have also an other question. Let $A$ be a symmetric matrix. If $\|A\|=0.01$ then there is an eigenvalue $\lambda$ with $|\lambda|\leq 0.01$, isn't it? We get that using the spectral radius: $\rho (A)=\max |\lambda_i|\leq \|A\|_1$, right? (Wondering)