Diagonal dominance versus positive semi-definiteness I know that for a symmetric matrix $A$, diagonal dominance, i.e. $$A_{ii} \ge \sum\limits_{j \ne i} |A_{ij}|$$ implies positive semi-definiteness. How about the other way? Does positive semi-definiteness imply diagonal dominance? Could you point to a proof or a counter example?

Quick counter example


>>> a=2*ones(3,3)+eye(3)
a =

3 2 2
2 3 2
2 2 3

>>> eig(a)
ans =

1.00000
1.00000
7.00000