Diagonal Dominance: a n ｴ n matrix A with real elements is dominant diagonal (dd) if there are n real numbers dj > 0, j = 1, 2, .., n such that
dj|ajj| > ・/font> iｹ j di|aij|
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| ｹ 0.
Theorem: If an n ｴ n matrix A is dominant diagonal and the diagonal is composed of negative elements (aii < 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.