Glossary

Diagonally dominant matrix

A matrix in which each diagonal element dominates the sum of the rest of its row.

Definition

It is strictly diagonally dominant by rows if ∣aii∣>∑j≠i∣aij∣|a_{ii}|>\sum_{j\neq i}|a_{ij}| for every row ii. This condition guarantees the matrix is invertible and that Jacobi and Gauss-Seidel converge from any starting point.

How it is used

It is the quick test before iterating: if dominance fails, reorder the equations to obtain it or check convergence via the spectral radius.