# Matrix index notation and Einstein summation

Can I ask why these two expressions are not equal?

\begin{align}A_{ij}V^j&\ne V^kA_{ki}\\A_{ij}B^{ij}&\ne A^i{}_jB^j{}_i\end{align}

• In both cases, it's because matrices are not in general symmetric.
– J.G.
Apr 8, 2022 at 20:46

First notice that tensors with two upper or two lower indices are not matrices, in that they represent linear transformations; they represent bilinear transformations. Tensors representing matrices have one upper and lower index.

Now in the first expression, if the tensor $$A$$ was symmetric, so that $$A_{ij} = A_{ji}$$, we would have:

$$A_{ij} V^j = A_{ji} V^j = V^j A_{ji} = V^k A_{ki}$$

Where in the last step we relabelled. Thus, in general, it is when the tensor is not symmetric that the two expressions are unequal.

Likewise if the tensor $$B$$ is symmetric, then we get:

$$A_{ij}B^{ij} = A_{ij} B^{ji} = A^i{}_j B^j{}_i$$

Hence, again in general, it is when either tensor $$A$$ or $$B$$ are not symmetric that the two expressions are unequal.