Perhaps this question is misguided but I am having difficulty writing a simple matrix expression in einstein notation.
In my expression I have a vector $v$ and I wish to define the scalar value, $a$, as the sum of the components of $v$, $v^i$. I know in einstein notation you use repeated indices to represent a summation but my problem is where would the second index appear? Do I have to use a kronecker delta with one index like so perhaps:
$$a=v^i\delta_i$$
Is this the usual convention?
Best Answer
The issue here is that the notion of the sum of the elements of a vector is not basis independent. So the way you would do this is to define a row-vector whose entries are all $1$s in the basis that you're working in, but note that this vector could look completely different in another basis.
I think $\delta_i$ is a logical name for this, but you might also see it called $1_i$. Then you're right, you can define the sum of the elements of $v$ as $v^i\delta_i$.