What Is the Kronecker Delta?

The Kronecker delta function, denoted δ_i,j, is a binary function that equals 1 if i and j are equal and equals 0 otherwise. Although it technically is a function of two variables, in practice it is used as notational shorthand, allowing complicated mathematical statements to be written compactly. Mathematicians, physicists and engineers who work in linear algebra, tensor analysis and digital signal processing use the Kronecker delta function as an expedient to convey in a single equation what might otherwise take several lines of text.

This function is most frequently employed to simplify the writing of equations that involve sigma notation, which is itself a concise method of referring to complicated sums. For example, if a company has 30 employees {e₁, e₂ ... e₃₀}, and each employee works a different number of hours {h₁, h₂ ... h₃₀} at a different hourly rate {r₁, r₂ ... r₃₀}, the total money paid to these employees for their work equals e₁*h₁*r₁ + e₂*h₂*r₂ + e₃*h₃*r₃ + ... e₃₀*h₃₀*r₃₀. Mathematicians can write this concisely as ∑_i e_i*h_i*r_i.

When describing physical systems that involve multiple dimensions, physicists frequently must use double summations. The practical scientific applications are very complex, but a concrete example shows how the Kronecker delta function can simplify expressions in these cases.

There are three clothing stores in a mall, each selling a different brand. A total of 20 styles of shirts are available: eight offered by store 1, seven offered by store 2 and five offered at store 3. Twelve styles of pants are available: five at store 1, three at store 2 and four at store 3. One can buy 240 possible outfits, because there are 20 options for the shirt and 12 options for the pants. Each combination yields a different outfit.

It is not as simple to calculate the number of ways to select an outfit in which the shirt and pants are from different stores. One can select a shirt from store 1 and pants from store 2 in 8*3 ways. There are 8*4 ways to select a shirt from store 1 and pants from store 3. Continuing in this manner, one finds the total number of outfits using articles from different stores is 8*3 + 8*4 + 7*5 + 7*4 + 5*5 + 5*3 = 199.

One could consider the availability of shirts and pants as two sequences, {s₁, s₂, s₃} = {8, 7, 5} and {p₁, p₂, p₃} = {5, 3, 4}. Then the Kronecker delta function allows this sum to be written as simply ∑_i ∑_j s_i * p_j * (1- δ_i,j). The (1- δ_i,j) term eliminates those outfits comprising a shirt and pants bought at the same store because in that case i = j, so δ_i,j = 1 and (1- δ_i,j) = 0. Multiplying the term by 0 removes it from the sum.

The Kronecker delta function is most frequently used when analyzing multidimensional spaces, but it also can be used when studying one-dimensional spaces, like the real number line. In that case, a single-input variant is often used: δ(n) = 1 if n = 0; δ(n) = 0 otherwise. To see how the Kronecker delta function can be used to simplify complex mathematical statements about the real numbers, one might consider the following two functions whose inputs are simplified fractions:

f(a/b) = a if a =b+1, f(a/b) = -b if b=a+1, and f(a/b) = 0 otherwise.
g(a/b) = a*δ(a-b-1) –b*δ(a-b+1)

The functions f and g are identical, but the definition for g is more compact and requires no English, so it can be understood by any mathematician in the world.

As illustrated by these examples, the inputs of the Kronecker delta function typically are integers that are connected to some sequence of values. The Dirac delta distribution is a continuous analog of the Kronecker delta function used when integrating functions rather than summing sequences.