Hadamard matrix ($H\in \{\pm 1{\}}^{n\times n},H{H}^T=nI$)

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 $H{H}^T=nI$ and $H/\sqrt{n}$ is an orthogonal matrix. In particular $H$ is invertible, with $H^{-1}=H^T/n$. The four-by-four exampleis symmetric and satisfies $H^2=4I$. It supplies an exact intertwiner in four-tile mixed-boundary transplantation between a disk and a nonorientable surface.