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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]], let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $p, q \\ge 0$ be integers.\nThe matrix $A$ is a *band matrix* with *lower bandwidth* $p$ and *upper bandwidth* $q$ when\n$$a_{ij} = 0 \\quad \\text{whenever } i \u003e j + p \\text{ or } j \u003e i + q.$$\nThe *bandwidth* of $A$ is often taken as the least $k$ such that the condition holds with $p = q = k$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 2 \u0026 0 \u0026 0 \\\\ 3 \u0026 4 \u0026 5 \u0026 0 \\\\ 6 \u0026 7 \u0026 8 \u0026 9 \\\\ 0 \u0026 1 \u0026 2 \u0026 3 \\end{pmatrix}$ has lower bandwidth $2$ and upper bandwidth $1$: the entries with $i \u003e j + 2$ or $j \u003e i + 1$ are all $0$.\nNon-example: $\\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 0 \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}$ has no upper bandwidth $q = 1$, because $a_{13} = 1$ and $3 \u003e 1 + 1$.\nWith $p = q = 0$ one gets the diagonal matrices, and with $p = q = 1$ the [[tridiagonal matrices|DARX1KZRHZ]].\nA square matrix with $p = 0$ is [[upper triangular|0N4XY7Z3NN]].\nThe term is mostly used when $p$ and $q$ are small compared with the size of the matrix.\n",
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