, a weighing matrix W
of order n
with weight w
is an n
-matrix such that
. A weighing matrix is also called a weighing design
For convenience, a weighing matrix of order
is often denoted by
A is equivalent to a conference matrix and a is an Hadamard matrix.
Some properties are immediate from the definition:
- The rows are pairwise orthogonal.
- Each row and each column has exactly non-zero elements.
- , since the definition means that (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.