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

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

where ${\displaystyle i}$ and ${\displaystyle j}$ take the values ${\displaystyle 1,2,\cdots ,n}$, and the inner product of vectors can be written as Here the Euclidean vectors are defined as n-tuples: ${\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}$ and ${\displaystyle \mathbf {b} =(b_{1},b_{2},...,b_{n})}$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over ${\displaystyle j}$.

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.

Properties

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.

Alternative notation

Using the Iverson bracket:

Often, a single-argument notation ${\displaystyle \delta _{i}}$ is used, which is equivalent to setting ${\displaystyle j=0}$:

In linear algebra, it can be thought of as a tensor, and is written ${\displaystyle \delta _{j}^{i}}$. Sometimes the Kronecker delta is called the substitution tensor.^[1]

Digital signal processing

In the study of digital signal processing (DSP), the unit sample function ${\displaystyle \mathrm{\delta <!--\; \delta \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Kronecker_delta_function
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