The Properties of the Kronecker Delta Function
Key Takeaways

The Kronecker delta function ij takes only two values, either 1 or 0.

The two indices i and j in the expression of the Kronecker delta function are interchangeable.

The mathematical statements involved with tensor analysis, linear algebra, and digital signal processing can be expressed as a single equation by using the Kronecker delta function.
The Kronecker delta function can be used to simplify complex mathematical expressions
It is impossible to explain theoretical physics without mentioning the Kronecker delta function. Most physicists, mathematicians, and engineers use the Kronecker delta function to express complex expressions. The Kronecker delta function is a powerful tensor that helps to compact and simplify long, complex expressions. The Kronecker delta function and the LeviCivita tensor are two of the most popular tensors in the technical domain. In this article, we will explore the Kronecker delta function and its properties.
The Kronecker Delta Function
In theoretical physics, physicists use the Kronecker delta function to express their ideas compactly and simply. The Kronecker delta function uses the lowercase greek letter with subscripts ‘i’ and ‘j’ and is expressed as δ_{ij}. The Kronecker delta function δ_{ij} takes only two values, either 1 or 0– which is why it is considered a binary function.
The Kronecker delta function yields either 1 or 0 depending on the two indices ‘i’ and ‘j’. The two indices are indicative of the dimension. For example, if we consider a threedimensional space, then the Kronecker delta function indices i and j can take values 1, 2, and 3.
The mathematical definition of the Kronecker delta function is:
In other words, the Kronecker delta function is equal to 1 when the indices i and j are equal. The Kronecker delta function yields a 0 value when the indices i and j are unequal.
Here are a few examples of the Kronecker delta function in the table below.
Kronecker Delta Function vs. Identity Matrix
Let’s distinguish between the identity matrix and the Kronecker Delta function.
Consider a 3 X 3 identity matrix:
The rows and columns are indicated by i and j, respectively. We can see that the matrix element I_{ij }= 1 when i and j are equal. Otherwise, the matrix element is zero.
From the mathematical equation given above, it is clear that any general element in the identity matrix can be written using the Kronecker delta function.
Note that n is the dimension of the identity matrix.
Kronecker Delta Function Rules
The Kronecker delta function is extensively used in mathematics, physics, and engineering. It is possible to simplify mathematical calculations involving the Kronecker delta function by applying certain rules of the function.

Symmetric Property
The two indices (i and j) in the expression of the Kronecker delta function are interchangeable.

Summation Property
In theoretical science, we may come up with products of Kronecker delta functions. If the product of the Kronecker delta function contains a common index, then it can be eliminated and can be rewritten using the rest of the indices.
Consider product δ_{ik}δ_{kj}. In this expression, both Kronecker delta functions contain the index ‘k’. The index ‘k’ can be deleted from the expression and the expression can be rewritten as
The summation index ‘k’ is contracted.

Contraction Property
Consider the product a_{j} δ_{jk}. Here, j is the common index in a_{j }and the Kronecker delta function. The common index ‘j’ can be deleted from the expression with the factor a_{j} getting the other index. The expression can be rewritten as

Cumulative Summation
Consider δ_{ii} where ‘i’ varies from 1 to n. When summation is performed over the index ‘i’, we get δ_{11} + δ_{22} + δ_{33} +………….δ_{nn}. According to the symmetric property, each Kronecker delta function yields 1 and gives the total sum equal to ‘n’.
δ_{jj }= n
Applying the Kronecker Delta Function in Mathematical Statements
The mathematical statements involved with tensor analysis, linear algebra, and digital signal processing can be expressed as a single equation with the Kronecker delta function. If you are a physicist, mathematician, or engineer, understanding the Kronecker delta function helps you define theoretical concepts mathematically in a much simpler way. The Kronecker delta function is used for analyzing problems in multidimensional spaces.
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