How to simplify Kronecker delta with einstein summation?

Question

I am trying to proof a vector identity. I have to prove the following;

I am bit confused how to simplify the following part.. $$\delta_{il} \delta_{jm} x_{j}y_{l}z_{m}$$

Any input is appreciated.

Answer 1

Remember Kronecker $\delta$ "is" the identity matrix, so contracting one suffix has the effect of replacing suffix. So you get $$\require{color} {\color{blue}\delta_{il}}{\color{red}\delta_{jm}x_j}{\color{blue}y_l}z_m={\color{red}x_m}{\color{blue}y_i}z_m=y_i(x\cdot z). $$