Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Question

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.

Answer 1

If $x_i = y_i$ for all $i = 1, \dots, n$, then $a_{ij} x^i y^j = a_{ij} y^i y^j = a_{ij}y^i x^j = a_{ij} x^j y^i.$

Answer 2

This equality can be written in terms of matrices as
$$x_i a_{ij} y_j=x^T A y=y_i a_{ij} x_j = y^T A x$$ where $x=(x_i),$ $y=(y_i)$, and $A=(a_{ij})$. If the vectors $x$ and $y$ are the same, then the above relation is identically true. As a side note, observe that $y^T A x= x^T A^T y$ and so the identity will hold for all $x,y$ if $A=A^T$ i.e. $A$ is symmetric.