Consider a diagonally dominant matrix $A$ with all positive entries. Is it true that the spectral radius of $A$ is lower than or equal to the maximum diagonal entry of $A$?
2026-03-27 02:39:13.1774579153
Bound spectral radius by diagonal elements of diagonally dominant matrix?
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No. Quite the contrary, the spectral radius is always greater than the largest diagonal element. Diagonal dominance is irrelevant here. Let $A$ be positive and $x$ be its Perron vector. By considering the $i$-th entry on both sides of $Ax=\rho(A)x$, we obtain $a_{ii}x_i<\sum_ja_{ij}x_j=\rho(A)x_i$. Hence $a_{ii}<\rho(A)$ for each $i$.