Hadamard matrices

Hadamard matrices are n × n

n × n matrices with elements -1 and +1, whose rows are orthogonal, so that m . Transpose[m]  n IdentityMatrix[n]

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m . Transpose[m]  n IdentityMatrix[n]. The matrices used in Walsh transforms are special cases with n = 2^s

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n=\!\(\*SuperscriptBox[\(2\),\(s\)]\). There are thought to be Hadamard matrices with every size n = 4k

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n = 4k (and for n > 2

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n > 2 no other sizes are possible); the number of distinct such matrices for each k up to 7 is 1, 1, 1, 5, 3, 60, 487. The so-called Paley family of Hadamard matrices for n = 4k = p + 1

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n = 4k = p + 1 with p prime are given by

PadLeft[Array[JacobiSymbol[#2 - #1, n - 1]&, {n, n} - 1] - IdentityMatrix[n - 1], {n, n}, 1]

PadLeft[Array[JacobiSymbol[#2 - #1, n - 1]&, {n, n} - 1] - IdentityMatrix[n - 1], {n, n}, 1]

Originally introduced by Jacques Hadamard in 1893 as the matrices with elements Abs[a] ≤ 1

Abs[a] ≤ 1 which attain the maximal possible determinant ± n^n/2

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±\!\(\*SuperscriptBox[\(n\),\(n/2\)]\), Hadamard matrices appear in various combinatorial problems, particularly design of exhaustive combinations of experiments and Reed–Muller error-correcting codes.