Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:

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

or with use of Iverson brackets:

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

For example, $\delta _{12}=0$ because $1\neq 2$ , whereas $\delta _{33}=1$ because $3=3$ .

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 linear algebra, the $n\times n$ identity matrix $\mathbf {I}$ has entries equal to the Kronecker delta:

I_{ij}=\delta _{ij}

where $i$ and $j$ take the values $1,2,\cdots ,n$ , and the inner product of vectors can be written as

\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: $\mathbf {a} =(a_{1},a_{2},\dots ,a_{n})$ and $\mathbf {b} =(b_{1},b_{2},...,b_{n})$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over $j$ .

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.

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