Skew Hadamard matrices are of special interest due to their uses, among others, in constructing orthogonal designs. Fast computational algorithms for skew Williamson-Hadamard transforms are constructed in this chapter. Fast algorithms of two groups of transforms based on skew-symmetric Williamson-Hadamard matrices are designed using the block structures of these matrices.
6.1 Skew Hadamard Matrices
Many constructions of Hadamard matrices are known, but not all of them give skew Hadamard matrices. In Ref. 1, the authors provide a survey on the existence and equivalence of skew-Hadamard matrices. In addition, they present some new skew Hadamard matrices of order 52 and improve the known lower bound on the number of the skew Hadamard matrices of this order. As of August 2006, skew Hadamard matrices were known to exist for all n ≤ 188 with n divisible by 4. The survey of known results about skew-Hadamard matrices is given in Ref. 33. It is conjectured that skew Hadamard matrices exist for n = 1, 2 and all n divisible by 4.
Definition 6.1.1: A matrix Am is called symmetric if ATm = Am. Matrix Am of order m is called skew symmetric if ATm = −Am. The following matrices are examples of skew-symmetric matrices of order 2, 3, and 4:
6.1.1 Properties of the skew-symmetric matrices
• If A = (ai,j) is a skew-symmetric matrix, then ai,j = 0.
• If A = (ai,j) is a skew-symmetric matrix, then ai,j = −ai,j.
• Sums and scalar products of skew-symmetric matrices are again skew symmetric, i.e., A and B are skew-symmetric matrices of the same orders, and c is the scalar number, then A + B and cA are skew-symmetric matrices.
• If A is a skew-symmetric matrix of order n, the determinant of A satisfies det(A) = det(AT) = det(−A) = (−1)n det(A).