Dominant Diagonal

Diagonal Dominance: a n ´ n matrix A with real elements is dominant diagonal (dd) if there are n real numbers d_j > 0, j = 1, 2, .., n such that

d_j|a_jj| > å _{i¹ j} d_i|a_ij|

for j = 1, 2, .., n.

There is two important theorems attached, both due to Lionel McKenzie (1960)

Theorem: If A is dominant diagonal, then |A| ¹ 0.

Theorem:  If an n ´ n matrix A is dominant diagonal and the diagonal is composed of negative elements (a_ii < 0 for all i = 1, .., n), then the real parts of all its eigenvalues are negative, i.e. A is a "stable matrix".

For proofs, see our mathematical section on stable matrices.