Hadamard-Matrices

[Annette.Warlich at t-online.de]

By matching a method from another area of math I came across
the problem of Hadamard-matrices, in the sense of the maximum
determinant problem.
In factor analysis I am experimenting with some rotation criteria -
say with a start of a matrix A to find some orthogonal rotation-
matrix, which maximizes (or minimizes) some criteria for that matrix.

Since Hadamard-Matrices H are orthogonal they can be rotated to a
n * I -matrix by

    H * T = n*I

so must a n*I-matrix be rotatable to H by

    H = n*I * T'

 From  factor-analysis the "quartimax"- and "varimax" rotation are known,
which maximize the variance of the squares of the entries of the columns.

One form some other descriptions of Hadamard-matrix besides the
maximum-determinant criterion is, that the variance of the squares
is zero.
Changing the Quartimax-criterion to a minimum criterion should
produce a Hadamard-matrix just by some simple canned rotation-
procedure of a factor-analysis-program.

something like
   n = 4
   A = sqrt(n) * Unit(n)
   H4 = rotate(A,"-quartimax")   // the *minimizing" criterion of quartimax

(With n=16 a certain precondition helped improving convergence.)

In fact, I just find the n=4,N=8,n=12 and n=16 - solutions by
using that criterion.
I was not able to detect such of higher order - already with the
n=16 case the convergence indicated the possibility of ending
in local minima.

I would like to understand this problem better - mainly: how can
the iterative process terminate without reaching its minimum.
I just tried it with higher numeric precision (up to 512 bits), but
it doesnt seem, that this is the problem.

Gottfried Helms