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  "body": "\n## Summary\n\nA square matrix is diagonally dominant when each row has a large diagonal entry.\nIn absolute value, it is at least the sum of the absolute values of the other entries of its row.\nIt is strictly diagonally dominant when this inequality is strict in every row.\nA strictly diagonally dominant matrix is invertible, by the [[Levy–Desplanques theorem|PWKN6ANA0G]].\nFor strictly diagonally dominant matrices, Gaussian elimination needs no pivoting, and the Jacobi and Gauss–Seidel iterations converge.\n\n## Explanations\n\n### L1 human\n\nIn a diagonally dominant matrix, each row is controlled by its diagonal entry: the other entries together cannot outweigh it.\nFor example, $\\begin{pmatrix} 4 \u0026 1 \u0026 -2 \\\\ 1 \u0026 3 \u0026 1 \\\\ 0 \u0026 2 \u0026 5 \\end{pmatrix}$ is diagonally dominant, since $4 \\ge 3$, $3 \\ge 2$ and $5 \\ge 2$.\nSuch a matrix behaves much like a diagonal matrix, which is easy to invert.\nThis makes it easy to see invertibility and to solve systems with it reliably.\n",
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