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Concept
Kronecker delta
Summary
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, because , whereas because .
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 identity matrix has entries equal to the Kronecker delta:
where and take the values , 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 .
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.
The following equations are satisfied:
Therefore, the matrix δ can be considered as an identity matrix.
Another useful representation is the following form:
This can be derived using the formula for the geometric series.
Using the Iverson bracket:
Often, a single-argument notation is used, which is equivalent to setting :
In linear algebra, it can be thought of as a tensor, and is written . Sometimes the Kronecker delta is called the substitution tensor.
In the study of digital signal processing (DSP), the unit sample function represents a special case of a 2-dimensional Kronecker delta function where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:
Or more generally where:
However, this is only a special case. In tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation does not exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero.
Official source
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