Hadamard Matrix - Generalizations and Special Cases

Generalizations and Special Cases

Many generalizations and special cases of Hadamard matrices have been investigated in the mathematical literature. One basic generalization is the weighing matrix, a square matrix in which entries may also be zero and which satisfies for some w, its weight. A weighing matrix with its weight equal to its order is a Hadamard matrix.

Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H*= n I_n where H* is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Butson-type Hadamard matrices are complex Hadamard matrices in which the entries are taken to be qth roots of unity. The term "complex Hadamard matrix" has been used by some authors to refer specifically to the case q = 4.

Regular Hadamard matrices are real Hadamard matrices whose row and column sums are all equal. A necessary condition on the existence of a regular n×n Hadamard matrix is that n be a perfect square. A circulant matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be of perfect square order. Moreover, if an n×n circulant Hadamard matrix existed with n>1 then n would necessarily have to be of the form 4u2 with u odd.

The circulant Hadamard matrix conjecture, however, asserts that, apart from the known 1×1 and 4×4 examples, no such matrices exist. This was verified for all but 26 values of u less than 104.