Diagonal dominance versus positive semi-definiteness

Question

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?

Answer 1

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

Answer 2

Take the following matrix:

$$ A=\left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right] $$

Notice that $A$ is positive semidefinite (it has a double eigenvalue at $\lambda=0$). But it is not diagonally dominant since $0=|A_{11}|<|A_{12}|=1$

Answer 3

A $2 \times 2$ counterexample is $\pmatrix{a^2 & a\cr a & 1\cr}$ for $|a| \ne 1$.