(ii) A symmetric diagonally dominant real matrix with nonnegative diagonal entries is positive semidefinite.
(iii) The multiplicity of the eigenvalue 0 of a symmetric real matrix with zero row sums and nonpositive off-diagonal entries equals the number of connected components of the graph defined on the index set of the rows and columns of , where two distinct indices are adjacent when .
Proof: Let be diagonally dominant, and let be an eigenvector, say, with . Let be maximal among the . Then . In all cases the result follows by comparing the absolute values of both sides.
In order to prove (i), assume that is singular, and that . Take absolute values on both sides. We find . Contradiction.
For (ii), assume that has a negative eigenvalue . Then . Contradiction.
For (iii), take again, and see how equality could hold everywhere in . We see that must be constant on the connected components of .