Constructing Hadamard matrices

A Hadamard matrix is an orthogonal matrix whose entires are all either 1 or − 1. For example is a Hadamard matrix of order 2. True to Stigler’s law of eponymy, James Joseph Sylvester investigated Hadamard matrices before Jacques Hadamard. Sylvester saw how to bootstrap the example above into more examples. If H is a Hadamard matrix, then the partitioned matrix Sylvester’s construction can be generalized as follows. If Hm is a Hadamard matrix of order m and Hn is a Hadamard matrix of order n, the the Kronecker product Hm⊗Hn is a Hadamard matrix of order mn. That is, you can form a new Hadamard matrix by taking the matrix Hm and replacing ±1 with the matrix ±Hn. Let S be the set of all possible Hadamard matrix orders. By the construction above, this set is closed under multiplication. Since 2 is in S, every power of 2 is in S. Hadamard proved that all n ≥ 4 in S are multiples of 4. That is, the condition 4 | n is necessary. He conjectured that it was also sufficient, though that has not…