Kronecker delta

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Kronecker delta (named after <a href="/facts/Leopold_Kronecker/T0lMQofb">Leopold Kronecker</a>) is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of two <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>, usually just non-negative <a href="/facts/Integer/PUwwrawx">integers</a>. The function is 1 if the variables are equal, and 0 otherwise:

δ
          
            i
            j
          
        
        =
        
          
            {
            
              
                
                  0
                
                
                  
                    if 
                  
                  i
                  ≠
                  j
                  ,
                
              
              
                
                  1
                
                
                  
                    if 
                  
                  i
                  =
                  j
                  .
                
              
            
            
          
        
      
    
    {\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}}

or with use of <a href="/facts/Iverson_bracket/HUuSwKe8">Iverson brackets</a>:

δ
          
            i
            j
          
        
        =
        [
        i
        =
        j
        ]
        
      
    
    {\displaystyle \delta _{ij}=[i=j]\,}

For example, 
 
 
 
 
 δ
 
 12
 
 
 =
 0
 
 
 {\displaystyle \delta _{12}=0}
 
 because 
 
 
 
 1
 ≠
 2
 
 
 {\displaystyle 1\neq 2}
 
, whereas 
 
 
 
 
 δ
 
 33
 
 
 =
 1
 
 
 {\displaystyle \delta _{33}=1}
 
 because 
 
 
 
 3
 =
 3
 
 
 {\displaystyle 3=3}
 
.
The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.
In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> 
 
 
 
 
 I
 
 
 
 {\displaystyle \mathbf {I} }
 
 has entries equal to the Kronecker delta:

I
          
            i
            j
          
        
        =
        
          δ
          
            i
            j
          
        
      
    
    {\displaystyle I_{ij}=\delta _{ij}}

where 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 j
 
 
 {\displaystyle j}
 
 take the values 
 
 
 
 1
 ,
 2
 ,
 ⋯
 ,
 n
 
 
 {\displaystyle 1,2,\cdots ,n}
 
, and the <a href="/facts/Inner_product/HoyjElEy">inner product</a> of <a href="/facts/Euclidean_vector/bCLrdksy">vectors</a> can be written as

a
        
        ⋅
        
          b
        
        =
        
          ∑
          
            i
            ,
            j
            =
            1
          
          
            n
          
        
        
          a
          
            i
          
        
        
          δ
          
            i
            j
          
        
        
          b
          
            j
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          a
          
            i
          
        
        
          b
          
            i
          
        
        .
      
    
    {\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i,j=1}^{n}a_{i}\delta _{ij}b_{j}=\sum _{i=1}^{n}a_{i}b_{i}.}

Here the <a href="/facts/Euclidean_vector/bCLrdksy">Euclidean vectors</a> are defined as n-tuples: 
 
 
 
 
 a
 
 =
 (
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}
 
 and 
 
 
 
 
 b
 
 =
 (
 
 b
 
 1
 
 
 ,
 
 b
 
 2
 
 
 ,
 .
 .
 .
 ,
 
 b
 
 n
 
 
 )
 
 
 {\displaystyle \mathbf {b} =(b_{1},b_{2},...,b_{n})}
 
 and the last step is obtained by using the values of the Kronecker delta to reduce the summation over 
 
 
 
 j
 
 
 {\displaystyle j}
 
.
It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.

Kronecker delta open-in-new

Kronecker delta