I'm working with an expression of the form:
$$x=(A\cdot B)y$$
Where $x$ and $y$ are column vectors of size $(n\times 1)$ and $A$ and $B$ are matrices of size $(n \times n)$ which I know represent transformations satisfying the definitions of a tensor. The dot, $\cdot$ represents the hadamard/element wise product. When I try to write this in ricci calculus notation I get the following:
$$x^i = A^i_jB^i_jy^j$$ $$=A^i_j(By)^i$$
However, now I have a $j$ which doesn't match any indices on the left hand side. I understand what I've done is wrong but I cant really see why and I'm not sure what the general rule is which I violated here.
Im worried that I don't know how to spot this kind of issue in bigger equations with more indices where its not so obvious I've done something wrong. What are the rules for performing tensor contractions when indices are repeated more than once?
Also does anyone know any good references with the rules/dos and don'ts for ricci notation clearly bullet pointed?