Matrix tensor indices

Question

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$

Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} \right )$$

or

$$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u^{i}\bigotimes u^{j} \right )\ ?$$

I'm inclined to believe the second form is correct. However, my notes stipulates the first one to be correct—quite possibly a blunder on the author's end.

Clarifications would be helpful.

Answer 1

This is a matter of convention that differs between literatures and depending on what you're doing. If you're going to use raised indices in a meaningful way, then I'd expect to see the basis written as $$ B = \{ u^i \}_{i=1}^{\infty} $$ with the index up on vectors. Then your second option for $K$ would be correct. Since you wrote the vectors in the basis with index down, it seems like you're in a convention where all indices are down and the first option is correct.

Answer 2

This depends very much on convention. If the basis is indexed by lower indices, the tensor product should be lower indices too. One could write the coefficients with upper indices and omit the summation sign like so: $$K = k^{ij}e_i\otimes e_j.$$ That would be the Einstein summation convention which is quite popular in mathematical physics. You can also read more about conventions with upper and lower indices here. In total the answer is "it depends" on the conventions made in the book you are reading, and also on the field you are working in. Mathematicians usually don't use summation convention.