The Kronecker delta function is a binary function that equals 1 if i and j are equal, otherwise 0. It simplifies writing equations involving sigma notation and is used in linear algebra, tensor analysis, and digital signal processing. It can also be used to simplify complex mathematical statements about the real numbers.
The Kronecker delta function, denoted δi,j, is a binary function that equals 1 if i and j are equal, otherwise it equals 0. Although technically a function of two variables, in practice it is used as a shorthand for notation, allowing you to compactly write complicated mathematical statements. Mathematicians, physicists, and engineers working in linear algebra, tensor analysis, and digital signal processing use the Kronecker delta function as a way to convey in a single equation what would otherwise require multiple lines of text.
This feature is most frequently used to simplify writing equations involving sigma notation, which is itself a concise method of referencing 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 is equal to e1*h1*r1 + e2*h2*r2 + e3*h3*r3 + … e30*h30*r30. Mathematicians can write this concisely as ∑i ei*hi*ri.
When describing physical systems involving multiple dimensions, physicists often have to use double sums. 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 shirt styles are available: eight from shop 1, seven from shop 2, and five from shop 3. Twelve pants styles are available: five from shop 1, three from shop 2, and four from shop 3. You can buy 240 possible outfits, because there are 20 shirt options and 12 pants options. Each combination produces a different outfit.
It’s not that easy to count the number of ways to select a suit where the shirt and pants come from different stores. You can select a shirt from shop 1 and pants from shop 2 in 8*3 ways. There are 8*4 ways to select a shirt from shop 1 and a pant from shop 3. Continuing this way, we find that the total number of garments using items from different shops is 8*3 + 8*4 + 7* 5 + 7 *4 + 5*5 + 5*3 = 199.
One could think of the availability of shirts and trousers as two sequences, {s1, s2, s3} = {8, 7, 5} and {p1, p2, p3} = {5, 3, 4}. Then the Kronecker delta function allows us to write this sum simply as ∑i ∑jsi * pj * (1- δi,j). The term (1- δi,j) eliminates those outfits composed of shirt and trousers bought in the same store because in that case i = j, therefore δi,j = 1 and (1- δi,j) = 0. Multiplying the term by 0 removes it from the sum.
The Kronecker delta function is used most frequently when studying multidimensional spaces, but it can also be used when studying one-dimensional spaces, such as the real number line. In this 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 could 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 ef(a/b) = 0 otherwise.g(a/b) = a *δ (ab-1) –b*δ(a-b+1)
The functions fe and g are identical, but the definition of g is more compact and does not require English, so it can be understood by any mathematician in the world.
As these examples illustrate, the inputs to the Kronecker delta function are usually integers connected to a sequence of values. The Dirac delta distribution is a continuous analog of the Kronecker delta function used when integrating functions rather than summing sequences.
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