On the Products of Hadamard Matrices, Williamson Matrices and Other Orthogonal Matrices using \(M\)-Structures

The new concept of \(M\)-structures is used to unify and generalize a number of concepts in Hadamard matrices, including Williamson matrices, Goethals-Seidel matrices, Wallis-Whiteman matrices, and generalized quaternion matrices. The concept is used to find many new symmetric Williamson-type matrices, both in sets of four and eight, and many new Hadamard matrices. We give as corollaries “that the existence of Hadamard matrices of orders \(4g\) and \(4h\) implies the existence of an Hadamard matrix of order \(8gh\)” and “the existence of Williamson type matrices of orders \(u\) and \(v\) implies the existence of Williamson type matrices of order \(2uv\)”. This work generalizes and utilizes the work of Masahiko Miyamoto and Mieko Yamada.