Weighing matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a weighing matrix of order 
 
 
 
 n
 
 
 {\displaystyle n}
 
 and weight 
 
 
 
 w
 
 
 {\displaystyle w}
 
 is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
 
 
 
 W
 
 
 {\displaystyle W}
 
 with entries from the set 
 
 
 
 {
 0
 ,
 1
 ,
 −
 1
 }
 
 
 {\displaystyle \{0,1,-1\}}
 
 such that:

W
        
          W
          
            
              T
            
          
        
        =
        w
        
          I
          
            n
          
        
      
    
    {\displaystyle WW^{\mathsf {T}}=wI_{n}}

Where 
 
 
 
 
 W
 
 
 T
 
 
 
 
 
 {\displaystyle W^{\mathsf {T}}}
 
 is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of 
 
 
 
 W
 
 
 {\displaystyle W}
 
 and 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
 is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> of order 
 
 
 
 n
 
 
 {\displaystyle n}
 
. The weight 
 
 
 
 w
 
 
 {\displaystyle w}
 
 is also called the degree of the matrix. For convenience, a weighing matrix of order 
 
 
 
 n
 
 
 {\displaystyle n}
 
 and weight 
 
 
 
 w
 
 
 {\displaystyle w}
 
 is often denoted by 
 
 
 
 W
 (
 n
 ,
 w
 )
 
 
 {\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 <a href="/facts/Balance_scale/6Yvv6frI">balance scale</a>, the <a href="/facts/Variance/ULBJKXD1">statistical variance</a> 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.

Weighing matrix open-in-new

Weighing matrix