A complex Hadamard matrix is any complex $N\times N$ matrix $H$ satisfying two conditions: $ |H_{jk}|=1$ for $j,k=1,2,...,N $ and $ HH^{\dagger}=N\mathbb{1}_N$ where $H^{\dagger}$ denotes the Hermitian transpose of $H$ and $\mathbb{1}_N$ is the identity matrix of a given dimension.
We denote the set of complex Hadamard matrices of size $N$ by $\mathcal{H}_N$.
Complex Hadamard matrices contain, as special cases, real Hadamard matrices and matrices of Butson type.
In physical literature the complex Hadamard matrices are sometimes called Zeilinger matrices [85] while in mathematical literature they are called unit Hadamard matrices (since all entries belong to the unit circle) [32] or generalized Hadamard matrices [61].