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Linear algebra is full of mysteries, with sometimes even single matrices hiding interesting mathematics, worth a lengthy contemplation. Well-known examples include the Pauli spin matrices, which are cult objects in physics, at the core of basic quantum mechanics, then the Dirac matrices, at the core of quantum electrodynamics (QED), and the Gell-Mann matrices, at the core of quantum chromodynamics (QCD).

This book is about a class of matrices which are particularly beautiful, no matter your aesthetics, and whose study is fun and pleasant, bringing us into lots of interesting mathematics, coming from algebra, geometry, analysis and probability. And which are of course potentially useful for something. These are the Hadamard matrices.

A complex Hadamard matrix is a square matrix [math]H\in M_N(\mathbb C)[/math] whose entries are on the unit circle in the complex plane, [math]|H_{ij}|=1[/math], and whose rows are pairwise orthogonal, with respect to the usual scalar product on [math]\mathbb C^N[/math]. The central example is the Fourier matrix, [math]F_N=(w^{ij})[/math] with [math]w=e^{2\pi iN}[/math], with the name coming from the fact that this is the matrix of the Fourier transform over the cyclic group [math]G=\mathbb Z_N[/math]. In general, a complex Hadamard matrix can be thought of as being a kind of “generalized Fourier matrix”, and the applications of the complex Hadamard matrices come from this.

There has been a lot of work on the Hadamard matrices, starting with Sylvester and Hadamard, long time ago, who looked at such matrices in the real case, [math]H\in M_N(\mathbb R)[/math]. Here the Hadamard matrix condition states that we must have [math]H\in M_N(\pm1)[/math], and that when comparing any two rows, the number of matchings must equal the number of mismatchings. The whole subject belongs to combinatorics, design theory and group theory, although there are some interesting analytic and probabilistic aspects as well.

Later on, it was realized that the general complex case, [math]H\in M_N(\mathbb C)[/math], is worth attention too, with motivation coming from discrete Fourier analysis, in a large sense. The subject here belongs to linear algebra, real algebraic geometry, combinatorics again, with plenty of constructions involving all sorts of tricky roots of unity, and with interesting analytic and probabilistic aspects as well. As for the potential applications, these belong to quantum physics, via constuctions involving operator algebras and quantum groups.

All in all, many things to be explained, and this book is an introduction to all this, with the aim of keeping things simple, but reasonably complete.

The first half of the book, Parts I and II, deals with the real Hadamard matrices, whose basic theory is quite elementary, and then with the basic theory in the complex case, using elementary algebraic and geometric techniques. Everything here is accessible with a minimal knowledge of linear algebra, and calculus in several variables.

The second half of the book, Parts III and IV, contains more advanced material, erring on the graduate side. We will discuss here advanced analytic techniques for dealing with the complex Hadamard matrices, and then we will have a look into potential applications to theoretical physics, at the level of quantum groups and operator algebras.

Although many things will be discussed in this book, this remains an introduction to the subject. There has been a huge amount of work in the real case, and we will discuss here only the very basic ideas behind this work. The same goes for the construction and classification work in the complex case, with once again a lot of literature waiting to be consulted, by the interested reader. As in what regards the applications, both in the real and the complex case, our discussion here will be something modest too, with the main aim being that of explaining the relation between the quantum groups and the Hadamard matrices, which is where the applications to quantum physics should come from.

There are several books dedicated to the Hadamard matrices, including Agaian , Horadam and Seberry-Yamada , all focusing on the real case, and by using algebraic methods. It is our hope that the present book can stand as a nice complement to these, written from a physicist's viewpoint, and as an invitation to the subject.

This book is partly based on a number of research papers that I wrote, and I am grateful to Julien Bichon, Ion Nechita and Jean-Marc Schlenker, for our joint work on the subject. Many thanks go as well to my cats, for advice with hunting techniques, martial arts, and more. When doing linear algebra, all this knowledge is very useful.

\ Cergy, July 2024

Teo Banica \baselineskip=15.95pt \tableofcontents \baselineskip=14pt

General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].