Permutation matrix ($P$)

A permutation matrix has exactly one entry equal to one in each row and column and zeros elsewhere. Left or right multiplication permutes coordinates, and $P^{-1}=P^T$.

For a permutation $\pi$ of an orthonormal basis, the weighted image columns are mutually orthogonal and have squared norms $|w_j{|}^2$. Hence $U^{†}U=\mathrm{diag}(|w_1{|}^2,\dots ,|w_d{|}^2)$ and the displayed criterion follows. All nonzero weights suffice for invertibility, but unitarity requires unit-modulus weights. This describes basis permutations decorated by phases without introducing unwanted changes of amplitude.

A signed permutation matrix has exactly one nonzero entry in each row and column, and every nonzero entry is $1$ or $-1$. It is a product of a diagonal sign matrix and a permutation matrix.