Curiosity about Kronecker's Delta?

Question

My professor gave this subject in the class (analytic geometry) and I thought it was very complicated, then I just decided to open Wikipedia entry on Kronecker's Delta and discovered it is quite simple. (With that I meant that I understand the function $\delta_{ij}$). I got curious with two things:

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• Why is it named after Kronecker? I mean, I tend to assume that results that are named after someone are considered hard/important. This delta seems to be simple - although I am aware that in mathematics, something that seems easy or trivial has deep consequences that only an experienced mathematician can see. If this is the case, I am not experienced.

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The Kronecker delta is used in many areas of mathematics, physics and engineering, primarily as an expedient to convey in a single equation what might otherwise take several lines of text.

• What is the meaning of "convey in a single equation what might otherwise take several lines of text"? I know that there might be some examples in the article I just mentioned, but I can't understand most of the article. Can you provide me an example of such an equation and it's big equivalent form?

Answer 1

It's nothing super-special, just a convenient shorthand. It does save typesetting though, because

$$\delta_{ij} = \left \{ \begin{array}{lr} 1 & i=j \\ 0 & i \neq j \end{array} \right.$$

which is a little bit much to write every time you use it. (Feel free to click edit to see how much that takes to write in LaTeX.)

Answer 2

Consider one example given in the Wikipedia page that you linked. It shows how to describe an identity matrix using the Kronecker's Delta function. If you attempted to describe this in words, it would be something like "a square matrix with one's on the diagonal and zeros everywhere else."

Answer 3

An example used routinely in physics is the orthonormality of a basis $\{e_1,e_2,\ldots\}$: $$\langle e_i | e_j \rangle = \delta_{ij}$$ This breaks down into two equations: $\langle e_i | e_j \rangle = 0$ when $i \neq j$ and $\langle e_i | e_i\rangle = 1$. While this may not seem to be much of a simplification, this allows one to simplify formulae containing sums of multiple inner products that occur often in physics.

As for actual mathematical depth, for a physicist, the Kronecker delta is simply a notational shorthand, a way of compactly writing sums and not much else.

You may be interested in this cool formula:

$$ \delta_{nm} = \frac{1}{2\pi} \int_0^{2\pi} e^{it(n-m)}\;\mathrm{d}t$$