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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe matrix $A$ is *tridiagonal* when $a_{ij} = 0$ for all $i, j$ with $\\abs{i - j} \u003e 1$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 2 \u0026 -1 \u0026 0 \\\\ -1 \u0026 2 \u0026 -1 \\\\ 0 \u0026 -1 \u0026 2 \\end{pmatrix}$ is tridiagonal, because the only entries with $\\abs{i - j} \u003e 1$ are $a_{13}$ and $a_{31}$, and both are $0$.\nNon-example: $\\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 0 \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}$ is not tridiagonal, because $a_{13} = 1$ and $\\abs{1 - 3} = 2$.\nA tridiagonal matrix is often written with diagonal $a_1, \\dots, a_n$, superdiagonal $b_1, \\dots, b_{n-1}$ and subdiagonal $c_1, \\dots, c_{n-1}$.\nThen the leading principal minors $f_k$ satisfy $f_k = a_k f_{k-1} - b_{k-1} c_{k-1} f_{k-2}$, with $f_0 = 1$ and $f_{-1} = 0$.\nEquivalently, $A$ is tridiagonal when it is both upper and lower [[Hessenberg|YCEEWJTX56]].\n",
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