Understanding the Kronecker Delta function

Homework Statement

I'm having some trouble understanding the Kronecker Delta function and how it is used. I understand the basics of it, if i=j, delta=1, if not, delta=0. However, I don't understand why:

$$\delta_{ii}=3$$
and
$$\delta_{ij}\delta_{ij}=3$$

Homework Equations

$$\delta_{ij}= \left\{\begin{array}{cc}1,&\mbox{ if }i=j,\\0, & \mbox{ if } i\neq k\end{array}\right.$$

The Attempt at a Solution

I have not been able to find any proofs using the Kronecker delta online anywhere so I'm not exactly sure of its function. My best guess of why $$\delta_{ii}=3$$ would be that you take the determinant of the identity function which would give you three. However, I'm not exactly sure what to do for the second one.

Also could someone explain why $$\delta_{ii}=\delta_{11}+\delta_{22}+\delta_{33}=1+1+1=3$$? That's what was in my book but they didn't have any explanation of why.

I don't understand why:

$$\delta_{ii}=3$$
and
$$\delta_{ij}\delta_{ij}=3$$

Where did you find these equalities? They are not correct. I think that they were meant to be these:

$$\sum_{i = 1}^{3} \delta_{ii}=3$$
and
$$\sum_{i = 1}^{3}\sum_{j = 1}^{3} \delta_{ij}=3$$

arildno
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The sigma summation sign is generally omitted in the Einstein summation convention, antonantal.

The reason is that the sign is entirely superfluous.

Thanks guys, that makes perfect sense now. I had forgotten that they could have been in Einstein notation.

The sigma summation sign is generally omitted in the Einstein summation convention, antonantal.

The reason is that the sign is entirely superfluous.

I knew about the Einstein notation but I didn't think it was used in the equalities above because the index variable appears only in the subscript position.

According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values.

arildno