Mathematical background

Let $H$ be the set of vector with components either -1 or 1 and the scalar matrices ${\bf A^{uv}}$, ${\bf u},{\bf v}\in H$,the set of matrices whose elements are

$$\vspace*{-0.1cm} A^{{\bf u}{\bf v}}_{ij}= \overline{a_{ij}}... ...= -1, \underline{a_{ij}}~{\rm if}~u_i.v_j= 1 \vspace*{-0.1cm}$$

The set of matrices ${\bf A^{uv}}$ has $2^{2n-1}$ elements. If all matrices in the set have a determinant of the same sign, then all the matrices A are regular [22].

Note than another simplification procedure may be built by using the spectral radius (see section 7.6).