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  "body": "\n## Summary\n\nThe Walsh–Hadamard transform takes a vector of length $2^m$ to the vector of its correlations with all $\\pm 1$ patterns given by a Sylvester [[Hadamard matrix|J0N4J606X0]].\nIt is the discrete Fourier transform of the group $(\\Z/2\\Z)^m$, and its rows are sampled [[Walsh functions|F12A5QKW8Z]].\nA fast algorithm computes it with about $m 2^m$ additions and subtractions, and no multiplications.\nIt is used in signal processing, coding theory, the analysis of Boolean functions and quantum computing.\n\n## Explanations\n\n### L1 human\n\nTake a list of $2^m$ numbers.\nFor each pattern of plus and minus signs from a fixed family of $2^m$ patterns, add up the numbers with those signs.\nThe resulting list of sums is the transform.\nFor $m = 1$ the patterns are $(+, +)$ and $(+, -)$, so $(a, b)$ goes to $(a + b, a - b)$.\nDoing the same step on pairs, then on pairs of pairs, and so on, gives the whole transform quickly.\n",
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