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!
2026-04-07 12:54:31.1775566471
operator norm and spectral radius
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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.