Been a while since GR, but I'm pretty sure some of those indices are supposed to be on top. A_{ij} = B_i^k * C_{jk} possibly, but don't quote me on that.
Since these are Euclidean spaces, the difference between contravariant and covariant indices is non-existent. So it doesn't matter whether the indices are upstairs or downstairs.
I would say not 'nonexistent', but rather 'avoidable'; or, even better, that there is an isomorphism in an appropriate sense. (Compare the situation for strings and lists of characters; the types are isomorphic, but I wouldn't say that the difference between them is non-existent!)
True. I'm just a lazy relativist though ;)
Yeah, but I find that -- like dimensional analysis -- keeping careful track of what's upstairs and downstairs can help me keep a computation from going off the rails.
As cabinpark (https://news.ycombinator.com/item?id=9165144) points out, it is not necessary to make this distinction; but, if you want to do so, canonically a matrix has one covariant and one contravariant index, so you would want something like
A_i^j = B_i^k*C_k^j .
Subscripts are how it's used in the more down to earth engineering applications like anisotropic thermal conductivity and elasticity. I can see why some authors might want to put the right-hand-side-only variables somewhere else for clarity though.