Weighing matrix

In mathematics, a weighing matrix W of order n with weight w is an n × n $(0,1,-1)$ -matrix such that $WW^{T}=wI$ . A weighing matrix is also called a weighing design. For convenience, a weighing matrix of order $n$ and weight $w$ is often denoted by $W(n,w)$ .

A $W(n,n-1)$ is equivalent to a conference matrix and a $W(n,n)$ is an Hadamard matrix.

Some properties are immediate from the definition:

The rows are pairwise orthogonal.
Each row and each column has exactly $w$ non-zero elements.
$W^{T}W=wI$ , since the definition means that $W^{-1} = w^{-1}W^{T}$ (assuming the weight is not 0).

Example of W(2, 2):

$\begin{pmatrix}-1 & 1 \\ 1 & 1\end{pmatrix}$

The main question about weighing matrices is their existence: for which values of n and w does there exist a W(n,w)? A great deal about this is unknown. An equally important but often overlooked question about weighing matrices is their enumeration: for a given n and w, how many W(n,w)'s are there? More deeply, one may ask for a classification in terms of structure, but this is far beyond our power at present, even for Hadamard or conference matrices.

External links

On Hotelling's Weighing Problem, Alexander M. Mood, Ann. Math. Statist. Volume 17, Number 4 (1946), 432-446.