diagonally dominant

A matrix where every row's diagonal entry outweighs, in absolute value, the sum of every other entry in that same row -- a simple, cost-free check that guarantees Jacobi and Gauss-Seidel both converge.

A matrix $A$ is strictly diagonally dominant if $\lvert a_{ii}\rvert>\sum_{j\ne i}\lvert a_{ij}\rvert$ for every row $i$ . This is a sufficient (not necessary) condition for both the Jacobi and Gauss-Seidel iterations to converge to the unique solution of $A\mathbf{x}=\mathbf{b}$ from any starting vector, since it forces the Jacobi iteration matrix's infinity norm below $1$ -- the same property Chapter 7, Section 2 already used to note that a diagonally dominant matrix never needs a pivot swap.

Examples

$\begin{pmatrix}4&1&-1\\2&6&1\\1&-1&5\end{pmatrix}$ is diagonally dominant: $4>1+1$ , $6>2+1$ , $5>1+1$ in every row.