Invitation to Hadamard matrices

An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. Such matrices appear in various contexts in combinatorics, and have applications to coding theory and its ramifications. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples of such matrices are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being generalized Fourier matrices, with applications to various questions in quantum physics. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.Comment: 400 page