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## Diagonally dominant matrices

A diagonally dominant matrix is a complex matrix with the property that we have for all . When all these inequalities are strict, the matrix is called strictly diagonally dominant.

Lemma 1.13.2 (i)   A strictly diagonally dominant complex matrix is nonsingular.

(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 .

Next: Support structure of eigenvectors Up: Gram matrices Previous: Gram matrices
Andries Brouwer 2003-09-30