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):

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.