Classification Results of Hadamard Matrices

In 1893 Hadamard proved that for any n x n matrix A over the complex numbers, with all of its entries of absolute value less than or equal to 1, it necessarily follows that

|det(A)| ≤ n^n/2 [n raised to the power n divided by two],

with equality if and only if the rows of A are mutually orthogonal and the absolute value of each entry is equal to 1 (See [2], [3]). Such matrices are now appropriately identified as Hadamard matrices, which provides an active area of research in both theoretical and applied fields of the sciences. In pure mathematics, Hadamard matrices are of interest due to their intrisic beauty as well as their applications to areas such as combinatorics, information theory, optics, operator algebras and quantum mechanics.

In this text we will introduce some fundamental properties of Hadamard matrices as well as provide a proofs of some classification results for real Hadamard matrices.