Kronecker delta in index notation

[Str1k3]

Homework Statement

what does the expression $$\delta_{ii}$$ mean?

Homework Equations

$$\delta_{ij}=1$$ if i = j and 0 otherwise

The Attempt at a Solution

What I'm not sure about is if both indices are in the subscript does this mean i can only use it on a term with a subscript or can it also act on a term with a superscript? what is the difference between this and $$\delta^{j}_{i}$$? and why can't it be used for index replacement?

 

[HallsofIvy]

The standard convention (often called the "Einstein summation convention" because Albert Einstein introduced it to simplify equations in his "General Theory of Relativity") is that when an index is repeated, it implies a sum over all possible values of that index.

Representing the Kroneker delta as a matrix, you get, in n dimensions, the n by n identity matrix. In that case $\delta_{ii}$ is the sum of the main diagonal (often called the "trace") and is equal to n.