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)":

:

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

* [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177730883 "On Hotelling's Weighing Problem"] , Alexander M. Mood, Ann. Math. Statist. Volume 17, Number 4 (1946), 432-446.