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# Regular sets of matrices and applications

## Abstract

SupposeA 1,...,A s are (1, - 1) matrices of order m satisfying

$$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$
(1)
$$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$
(2)
$$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m }$$
(3)
$$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$
(4)

Call A1,…,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

1. (i)

if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm

2. (ii)

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

3. (iii)

if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists

4. (iv)

an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h

5. (v)

Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n

6. (vi)

anOD(4mp;ms1,…,msu whenever anOD (4p;s1,…,su)exists and s = 2p

7. (vii)

a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 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.

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## Author information

Correspondence to Jennifer Seberry or Xian-Mo Zhang.

## Rights and permissions

Reprints and Permissions

Seberry, J., Zhang, X. Regular sets of matrices and applications. Graphs and Combinatorics 9, 185 (1993). https://doi.org/10.1007/BF02988305