Weighing matrix

In mathematics, a weighing matrix of order ${\displaystyle n}$ and weight ${\displaystyle w}$ is a matrix ${\displaystyle W}$ with entries from the set ${\displaystyle \{0,1,-1\}}$ such that:

${\displaystyle WW^{\mathsf {T}}=wI_{n}}$

Where ${\displaystyle W^{\mathsf {T}}}$ is the transpose of ${\displaystyle W}$ and ${\displaystyle I_{n}}$ is the identity matrix of order ${\displaystyle n}$. The weight ${\displaystyle w}$ is also called the degree of the matrix. For convenience, a weighing matrix of order ${\displaystyle n}$ and weight ${\displaystyle w}$ is often denoted by ${\displaystyle W(n,w)}$.^[3]

Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.^[1][2]