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# Weighing matrix

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

A $W\left(n,n-1\right)$ is equivalent to a conference matrix and a $W\left(n,n\right)$ 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.
• $W^\left\{T\right\}W=wI$, since the definition means that $W^\left\{-1\right\} = w^\left\{-1\right\}W^\left\{T\right\}$ (assuming the weight is not 0).

Example of W(2, 2):

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

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.