Do the free indices referring to the same dimension have to be denoted by the same letter on both sides of an expression when using Einstein notation?

Question

If there is an equation in vector form, ie:

$$y=Ax$$

I know that I can rewrite this using Einstein notation:

$$y_i = A_{ij}x_j$$

I believe the above is valid. Would this:

$$y_i = A_{ji}x_i$$

be still valid though, and does this mean the same thing?

I think it would be reasonable if it was valid, since there are only one free index on both sides of the equation, so there is no ambiguity. I do not however see this in texts, so I think this is in fact invalid.

So, is $y_i = A_{ji}x_i$ a valid expression?

Answer 1

Yes, free indices corresponding to equal quantities should use the same letter.

$y_i$ means the "$i$th entry of $y$". $A_{ji}x_i$, on the other hand, is the $j$th entry of the matrix product of $Ax$. So, the equation $y_i = A_{ji}x_i$ amounts to saying that the $i$th entry of $y$ is equal to the $j$th entry of $Ax$ for arbitrary indices $i,j$.

This is different from saying that the $i$th entry of $y$ is equal to the $i$th entry of $Ax$ for arbitrary indices $i$, which is what is conventionally meant by the expression $y = Ax$.