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:

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 ${\displaystyle W(n,w)}$.

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.

Some properties are immediate from the definition. If ${\displaystyle W}$ is a ${\displaystyle W(n,w)}$, then:

A weighing matrix is a generalization of a Hadamard matrix, which does not allow zero entries. As two special cases, a ${\displaystyle W(n,n)}$ is a Hadamard matrix and a ${\displaystyle W(n,n-1)}$ is equivalent to a conference matrix.

Weighing matrices take their name from the problem of measuring the weight of multiple objects. If a measuring device has a statistical variance of ${\displaystyle \sigma ^{2}}$, then measuring the weights of ${\displaystyle N}$ objects and subtracting the (equally imprecise) tare weight will result in a final measurement with a variance of ${\displaystyle 2\sigma ^{2}}$. It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a balance scale where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.

An order ${\displaystyle n}$ matrix ${\displaystyle W}$ can be used to represent the placement of ${\displaystyle n}$ objects—including the tare weight—in ${\displaystyle n}$ trials. Suppose the left pan of the balance scale adds to the measurement and the right pan subtracts from the measurement. Each element of this matrix ${\displaystyle w_{ij}}$ will have:

Let ${\displaystyle \mathbf {y} }$ be a column vector of the measurements of each of the ${\displaystyle n}$ trials, let ${\displaystyle \mathbf {\epsilon } }$ be the errors to these measurements each independent and identically distributed with variance ${\displaystyle \sigma ^{2}}$, and let ${\displaystyle \mathbf {\theta } }$ be a column vector of the true weights of each of the ${\displaystyle n}$ objects. Then we have: