Proportion of 1s in a Hadamard matrix

The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product.

Starting with a Hadamard matrix H₀ and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by

H_n+1 = G ⊗ H_n

for  positive integers n. This is known as the generalized Sylvester method.

Let p_n be the proportion of 1s in H_n and let q be the proportion of 1s in G. Then you can show that the recurrence holds

p_n+1 = q p_n + (1 − q)(1 − p_n).

You can solve the recurrence to show that

lim_{n → ∞} p_n = ½

and so as the iterations proceed, the ratio of number of 1s to the number of −1s approaches 1.

This doesn’t say anything Hadamard matrices in general, but it does apply to all Hadamard matrices created by repeatedly applying the generalized Sylvester method.

If you set G and H equal to the matrix

then p₀ = q = ¾. Then for n = 1, 2, 3, …, 8 the values of p_n are

0.625
0.5625
0.53125
0.515625
0.5078125
0.50390625
0.501953125
0.5009765625.