#### Abstract

Suppose A1,....,As are (1, -1) matrices of order m satisfying

A_{i}A_{j}=J, i,jє{1,...,s}

A^{t}_{i}A_{j}=A^{t}_{j}A_{i}=J, i≠j, i,jє{1,...,s}

∑(A_{i}A^{t}_{i }+ A^{T}_{i}Ai) = 2smI_{m}

JA_{i }= A_{i}J = aJ, i є {1,....,s}, a constant

Call A_{1},.....,A_{s} a regular s-set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in "New Hadamard matrices and conference matrices obtained via Mathon's construction". Graphs and Combinatorics. 4(1988), 355-377. In this paper, we prove that

(i) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular s-set of order mn when t = sm

(ii) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular s-set of order mn when 2t = sm (m is odd)

( iii) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular 2s-set of order mn when t = 2sm As applications, we prove that if there exist a regular s-set of order m there exists

(iv) an Hadamard matrices of order 4hm whenever there exists an Hadamard matrix of order 4h and s = 2h

(v) Williamson type matrices of order nm whenever there exists Williamson type matrices of order n and s = 2n

(vi) an OD(4mp;ms_{1}...., ms_{u}) whenever an OD(4p;S_{1},...,s_{u}) exists and s = 2p

(vii) a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c and s = 2c This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.

## Publication Details

Jennifer Seberry and Xian-Mo Zhang, Regular sets of matrices and applications, Graphs and Combinatorics, 9, (1993), 185-195.