I have this question in my homework:
I am not familiar with using kronecker delta or index notation to solve problem. I would like to ask if I have used the hint to express M and inverse of M, is it valid to do (q+w)(e+r)=qe+qr+we+wr just like algebra? or how to solve this question?
Best Answer
Hint: Note that $$ (c \delta_{ij} + d z_i z_j)(a \delta_{jk} + b z_j z_k)=\\ ac \; \delta_{ij}\delta_{jk} + bc z_j z_k \;\delta_{ij} + ad z_iz_j\;\delta_{jk} + bd\;z_iz_j^2z_k $$ We then have (with the implied Einstein notation) $$ (M^{-1})_{ij}M_{jk} = \\ \sum_{j=1}^n (ac \; \delta_{ij}\delta_{jk} + bc z_j z_k \;\delta_{ij} + ad z_iz_j\;\delta_{jk} + bd\;z_iz_j^2z_k) =\\ ac \;\delta_{jk} + (bc + ad)z_iz_k + bd z_iz_k\|z\|^2 $$