Weighing matrix

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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^TW = wI, since the definition means that W ^{− 1} = w ^{− 1}W^T (assuming the weight is not 0).

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.