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 Levi-Civita 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 three-dimensional 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. 

1. Symmetric Property

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

2.  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_ikδ_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.

3. 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

4. Cumulative Summation

Consider δ_ii where ‘i’ varies from 1 to n. When summation is performed over the index ‘i’, we get δ₁₁ + δ₂₂ + δ₃₃ +………….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 multi-dimensional spaces.

Cadence’s CFD software can assist you in simplifying complex problems involving multi-dimensional spaces. Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you. 

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