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:
![{\displaystyle \delta _{ij}=[i=j]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1419db152cf3420e549896141fbf42846adb3141)
where the Kronecker delta δij is a piecewise function of variables i and j. For example, δ1 2 = 0, whereas δ3 3 = 1.
The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.
In linear algebra, the n × n identity matrix I has entries equal to the Kronecker delta:
where i and j take the values 1, 2, ..., n, and the inner product of vectors can be written as
Here the Euclidean vectors are defined as n-tuples: and and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j.
The restriction to positive or non-negative integers is common, but in fact, the Kronecker delta can be defined on an arbitrary set.
https://en.wikipedia.org/wiki/Kronecker_delta
No comments:
Post a Comment