1
$\begingroup$

Can I ask why these two expressions are not equal?

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

$\endgroup$
1
  • 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

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.