**Dominant Diagonal**

Diagonal Dominance: a n ｴ n matrixAwith real elements is dominant diagonal (dd) if there are n real numbers d_{j }> 0, j = 1, 2, .., n such thatd

_{j}|a_{jj}| > ・/font>_{iｹ j}d_{i}|a_{ij}|for j = 1, 2, .., n.

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

Theorem: IfAis dominant diagonal, then |A| ｹ 0.

Theorem: If an n ｴ n matrixAis 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.Ais a "stable matrix".

For proofs, see our mathematical section on stable matrices.