A Life ' s Work on Hadamard Matrices ,

1 Hadamard matrices in Space Communications One hundred years ago, in 1893, Jacques Hadamard 21] found square matrices of orders 12 and 20, with entries 1, which had all their rows (and columns) orthogonal. These matrices, X = (x ij), satissed the equality of the following inequality jdet Xj 2 n i=1 n X j=1 jx ij j 2 and had maximal determinant. Hadamard actually asked the question of matrices with entries on the unit disc but his name has become associated with the real matrices. Hadamard was not the rst to study these matrices for J.J. Sylvester in 1857 in his seminal paper \Thoughts on inverse orthogonal matrices, simultaneous sign-successions and tessellated pavements in two or more colours with application to Newton's rule, ornamental tile work and the theory of numbers" 55] had found such matrices for all orders which are powers of two. Nevertheless, Hadamard showed matrices with elements 1 and maximal determinant could exist for all orders 1, 2, and 4t and so the Hadamard conjecture \that there exists an Hadamard matrix, or square matrix with every element 1 and all row (column) vectors orthogonal" came from here. The survey by J. Seberry and M. Yamada 51] indicates the progress that has been made in the past 100 years. Hadamard's inequality applies to matrices from the unit circle and matrices with entries 1; i and pairwise orthogonal rows (and columns) are called complex Hadamard matrices (note the scalar product is a:b = P a i b i for complex numbers). These matrices were rst studied by R.J. Turyn 58]. We believe complex Hadamard matrices exist for every order n 0 (mod 2).