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  "body": "\n## Summary\n\nA Toeplitz matrix is a matrix that is constant along each diagonal parallel to the main diagonal.\nIts entry in row $i$ and column $j$ depends only on the difference $i - j$.\nToeplitz matrices describe discrete convolutions and stationary time series, and linear systems with them can be solved faster than general systems.\nThey are closely related to [[Hankel matrices|PQ1D8FQB3E]], which are constant along the anti-diagonals.\n\n## Explanations\n\n### L1 human\n\nA Toeplitz matrix is fixed by its first row and its first column; every other entry copies the entry up and to the left of it.\nFor example, $\\begin{pmatrix} a \u0026 b \u0026 c \\\\ d \u0026 a \u0026 b \\\\ e \u0026 d \u0026 a \\end{pmatrix}$ is Toeplitz.\nSuch matrices come up whenever a rule acts the same way at every position, like a moving average over a list of numbers.\nSo an $n \\times n$ Toeplitz matrix is described by only $2n - 1$ numbers instead of $n^2$.\n",
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