Two tensors of different ranks can be multiplied in two ways:
- Outer product: Like $A^{ijk}_{lm}B^{r}_s=C^{ijkr}_{lms}$ is the outer product of $A^{ijk}_{lm}$ and $B^{r}_s$
- Inner product: (outer product of two tensors followed by a contraction) $A^{ij}_k$ and $B^{l}_{mn}$ is $A^{ij}_kB^{l}_{mn}\stackrel{k=l}{=}A^{ij}_lB^{l}_{mn}$
contraction: If in a mixed tensor, we put one contravariant and one covariant index equal, and the sum is performed over that repeated index, then a new tensor of the lower order is obtained. $A^{ijk}_{pq}\stackrel{q=k}{=}B^{ij}_p\stackrel{p=j}{=}C^i$
I am confused about these products. Why is such a product needed in tensor, or how is it related to our usual product (from inner product space)?
The Outer product makes some sense, but I couldn't make sense of the inner products. Like why here contraction was introduced without any context?