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 matrixH₀ 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 =

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 setG andH equal to the matrix

thenp₀ =q = 3/4. Then forn = 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.

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