For any real-valued square matrix with all positive entries, by Perron-Frobenius theory, we have that the matrix has a dominant eigenvalue that is real, positive, and of multiplicity 1. Thus, the spectral radius is equal to the largest eigenvalue.
Q: Is the operator norm for such a matrix always equal to the spectral radius, or can it be strictly larger?
Best Answer
The operator norm of a positive matrix can be larger than the spectral radius. E.g. when $A=uv^T$ for some two non-parallel positive vectors $u$ and $v$, we have $$ \|A\|_2=\|u\|_2\|v\|_2>v^Tu=\rho(A). $$ We do know a few things about the relationship between $\|A\|_2$ and $\rho(A)$: