= Hadamard matrix
{c}
{title2=$H\in\{\pm1\}^{n\times n},\ HH^T=nI$}
{wiki}
A <Hadamard matrix> is a real square <matrix> with entries $+1$ and $-1$ whose rows are pairwise <orthogonal>. Each row has squared <norm> $n$, so $HH^T=nI$ and $H/\sqrt n$ is an <orthogonal matrix>. In particular $H$ is invertible, with $H^{-1}=H^T/n$. The four-by-four example
$$
H=\begin{pmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{pmatrix}
$$
is symmetric and satisfies $H^2=4I$. It supplies an exact intertwiner in <four-tile mixed-boundary transplantation between a disk and a nonorientable surface>.
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