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is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1?

many thanks for any help, it is much appreciated!

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No. Consider for example $$ \begin{bmatrix}0&2\\0&0\end{bmatrix}. $$ Its spectral radius is $0$, and its operator norm is $2$.

Now, for normal operators, the operator norm and the spectral radius agree.

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  • $\begingroup$ You are welcome :) $\endgroup$ – Martin Argerami Nov 22 '12 at 4:41

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