Can I ask why these two expressions are not equal?

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

  • 3
    $\begingroup$ In both cases, it's because matrices are not in general symmetric. $\endgroup$
    – J.G.
    Apr 8, 2022 at 20:46

1 Answer 1


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.


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