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  "body": "\n## Definition\n\nLet $A = (a_{nk})_{n, k \\ge 0}$ be an infinite matrix of complex numbers.\nFor a complex [[sequence|0QE8STGJH2]] $s = (s_k)_{k \\ge 0}$, the *$A$-transform* of $s$ is the sequence\n$$(As)_n = \\sum_{k=0}^{\\infty} a_{nk} s_k \\qquad (n \\ge 0),$$\ndefined when each of these [[series|0JH4MN4K1C]] converges.\nThe sequence $s$ is *$A$-summable to $L$* if $(As)_n$ is defined for every $n$ and $(As)_n \\to L$ as $n \\to \\infty$.\nThe resulting [[summability method|X80DZEXQDQ]], with $D_A$ the set of $A$-summable sequences and $A(s) = \\lim_n (As)_n$, is the *matrix method* given by $A$.\n\n## Remarks\n\nExample: the Cesàro matrix has $a_{nk} = \\frac{1}{n+1}$ for $k \\le n$ and $a_{nk} = 0$ for $k \u003e n$.\nFor $s = (1, 0, 1, 0, \\dots)$ one has $(As)_n = \\frac{\\lceil (n+1)/2 \\rceil}{n+1} \\to \\frac12$, so $s$ is $A$-summable to $\\frac12$.\nNon-example: for $a_{nk} = 2^{-k-1}$ and $s_k = 2^k$, the series $\\sum_k a_{nk} s_k = \\sum_k \\frac12$ diverges, so $(As)_n$ is not defined and $s$ is not $A$-summable.\nThe identity matrix gives ordinary convergence.\nEvery matrix method is linear, because $A(\\lambda s + \\mu t) = \\lambda As + \\mu At$.\n\n## Explanations\n\n### L1 human\n\nMultiply the infinite matrix by the sequence, viewed as an infinite column, and take the limit of the result.\n",
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