Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker), is a function of two variables, usually non-negative integers. 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 Iverson brackets:

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.

Generalized versions of the Kronecker delta have found applications in differential geometry and modern tensor calculus, particularly in formulations of gauge theory and topological field models.

In linear algebra, the

n

×

n

{\displaystyle n\times n}

identity matrix

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 inner product of vectors 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 Euclidean vectors 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}

.