Given $m$ data points $x^i,i=1,$dots,$m,K-$means clustering algorithm groups them into $k$ clusters by mini$-$

mizing the distortion function over $\{r^{ij},{}^j\}$

$J={\displaystyle \underset{i=1}{\overset{m}{}}}{\displaystyle \underset{j=1}{\overset{k}{}}}{r}^{ij}||x^i-{}^j|{|}^2,$

where $r^{ij}=1$ if $x^i$ belongs to the $j-$th cluster and $r^{ij}=0$ otherwise.

Derive mathematically what should be the assignment variables $r^{ij}$ be to minimize the

distortion function $J,$ when the centroids ${}^j$ are fixed.