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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nA Hadamard matrix of order $n$ is an $n \\times n$ [[matrix|0N1MY1T0W9]] $H$ with entries in $\\set{1, -1}$ such that $H H^T = n I_n$.\nHere $H^T$ is the [[transpose|0W809K6AW0]] of $H$ and $I_n$ is the [[identity matrix|06DW9BAWB1]] of size $n$.\n\n## Remarks\n\nThe condition says that any two distinct rows have dot product $0$, so they agree in exactly $n/2$ places.\nSince $H^{-1} = H^T / n$, also $H^T H = n I_n$, so the columns are pairwise orthogonal as well.\nExample: $H_2 = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 -1 \\end{pmatrix}$ satisfies $H_2 H_2^T = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 2 \\end{pmatrix} = 2 I_2$.\nThe [[Kronecker product|0JK6M18QZH]] of two Hadamard matrices is a Hadamard matrix, so the Kronecker powers of $H_2$ give Hadamard matrices of every order $2^k$.\nNon-example: no $3 \\times 3$ matrix with entries $\\pm 1$ is Hadamard, since the dot product of two rows is a sum of three terms $\\pm 1$, which is odd and so not $0$.\nA Hadamard matrix of order $n$ can exist only if $n = 1$, $n = 2$ or $n$ is a multiple of $4$; the [[Hadamard conjecture|6X9S10E3M2]] asks whether all multiples of $4$ occur.\nFor such $H$, $\\abs{\\det H} = n^{n/2}$.\n",
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