We have seen in Chapter 1 that Hadamard's original construction of Hadamard matrices states that the Kronecker product of Hadamard matrices of orders m and n is a Hadamard matrix of order mn. The multiplicative theorem was proposed in 1981 by Agaian and Sarukhanyan (see also Ref. 2). They demonstrated how to multiply Williamson-Hadamard matrices in order to obtain aWilliamson-Hadamard matrix of order mn/2. This result has been extended by the following:

• Craigen et al. show how to multiply four Hadamard matrices of orders m, n, p, q in order to obtain a Hadamard matrix of order mnpq/16.

• Agaian and Sarukhanyan et al. show how to multiply several Hadamard matrices of orders n_i, i = 1, 2, . . . , k + 1, to obtain a Hadamard matrix of order (n₁n₂ . . . n_k+1)/2k, k = 1, 2, . . . . They obtained a similar result for A(n,k)-type Hadamard matrices and for Baumert-Hall, Plotkin, and Goethals-Seidel arrays.

• Seberry and Yamada investigated the multiplicative theorem of Hadamard matrices of the generalized quaternion type using the M-structure.

• Phoong and Chang show that the Agaian and Sarukhanyan theorem results can be generalized to the case of antipodal paraunitary (APU) matrices. A matrix function H(z) is said to be paraunitary (PU) if it is unitary for all values of the parameters z, H(z)H^T (1/z) = nI_nn ≥ 2. One attractive feature of these matrices is their energy preservation properties that can reduce the noise or error amplification problem. For further details of PU matrices and their applications, we refer the reader to Refs. 8-10. A PU matrix is said to be an APU matrix if all of its coefficient matrices have ±1 as their entries. For the special case of constant (memoryless) matrices, APU matrices reduce to the well-known Hadamard matrices.