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  "body": "\n## Definition\n\nLet $A = (a_{ij})$ be an $n \\times n$ complex [[square matrix|0JXZKX526R]].\nThe matrix $A$ is *diagonally dominant* (by rows) when\n$$\\abs{a_{ii}} \\ge \\sum_{j \\ne i} \\abs{a_{ij}} \\qquad \\text{for all } i = 1, \\dots, n.$$\nIt is *strictly diagonally dominant* when this inequality is strict for all $i$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 3 \u0026 -1 \\\\ 1 \u0026 2 \\end{pmatrix}$ is strictly diagonally dominant, because $3 \u003e 1$ and $2 \u003e 1$.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 3 \\end{pmatrix}$ is not diagonally dominant, because in the first row $\\abs{1} \u003c \\abs{2}$.\nDiagonal dominance by columns is the same condition for the transpose $A^T$.\nA strictly diagonally dominant matrix is invertible, by the [[Levy–Desplanques theorem|PWKN6ANA0G]].\nA diagonally dominant matrix that is not strictly so can be singular, for example $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$.\n",
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