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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$.\nThe matrix $A$ is a *Toeplitz matrix* when $a_{ij} = a_{i+1, j+1}$ for all $1 \\le i \u003c m$ and $1 \\le j \u003c n$.\nEquivalently, there are elements $t_k \\in R$, for $1 - n \\le k \\le m - 1$, with $a_{ij} = t_{i-j}$ for all $i, j$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 1 \u0026 2 \\\\ 5 \u0026 4 \u0026 1 \\end{pmatrix}$ is Toeplitz, with $t_0 = 1$, $t_{-1} = 2$, $t_{-2} = 3$, $t_1 = 4$, $t_2 = 5$.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$ is not Toeplitz, because $a_{11} = 1 \\ne 4 = a_{22}$.\nFor $m = n$, the circulant matrices are the Toeplitz matrices with $t_k = t_{k-n}$ for $1 \\le k \\le n-1$.\nIf $J$ is the $m \\times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Toeplitz exactly when $JA$ is a [[Hankel matrix|PQ1D8FQB3E]].\n",
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