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:

or with use of Iverson brackets:

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

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

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}.

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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