Trying to understand the sigma expansion in an answer Iteration 2: Reassign cluster labels according to the smallest squared distances:
$\begin{array}{ccccc}\text { Data point } X_{i} & \left\|X_{i}-(0,1)\right\|^{2} & \left\|X_{i}-(4,4)\right\|^{2} & \left\|X_{i}-(6,1)\right\|^{2} & \text { Cluster Label } \\ (6,-1) & 40 & 29 & 4 & 3 \\ (4,3) & 20 & 1 & 8 & 2 \\ (6,3) & 40 & 5 & 4 & 3 \\ (1,1) & 1 & 18 & 25 & 1 \\ (4,5) & 32 & 1 & 20 & 2 \\ (-1,1) & 1 & 34 & 49 & 1\end{array}$
$$
\begin{aligned}
&\left\{C_{l_{1}}, C_{l_{2}}, C_{l_{3}}\right\}=\{\{(1,1),(-1,1)\},\{(4,3),(4,5)\},\{(6,-1),(6,3)\}\} \\
&\left\{\mu_{l_{1}}, \mu_{l_{2}}, \mu_{l_{3}}\right\}=\{(0,1),(4,4),(6,1)\}
\end{aligned}
$$
The cost function which is the sum of the distances of the data points from each cluster to their centroids is given by
$$
\begin{aligned}
C\left(l_{1}, l_{2}, l_{3}, \mu_{l_{1}}, \mu_{l_{2}}, \mu_{l_{3}}\right)=& \frac{1}{6} \sum_{l_{j}=1}^{K=3} \sum_{i=1}^{N=6}\left\|X_{i}-\mu_{l_{j}}\right\|^{2} \\
=& \frac{1}{6}\left\{\left\|X_{1}-\mu_{l_{3}}\right\|^{2}+\left\|X_{2}-\mu_{l_{2}}\right\|^{2}+\left\|X_{3}-\mu_{l_{3}}\right\|^{2}\right.\\
&\left.+\left\|X_{4}-\mu_{l_{1}}\right\|^{2}+\left\|X_{5}-\mu_{l_{2}}\right\|^{2}+\left\|X_{6}-\mu_{l_{1}}\right\|^{2}\right\} \\
=& \frac{1}{6}\{4+1+4+1+1+1\}=\frac{12}{6}=2
\end{aligned}
$$
Hi, I was reading a solution. But I didn't understand how the author find this sigma expansion, that is I didn't understand the following:How did get the author these, can you explain?
$\begin{aligned}=& \frac{1}{6}\left\{\left\|X_{1}-\mu_{l_{3}}\right\|^{2}+\left\|X_{2}-\mu_{l_{2}}\right\|^{2}+\left\|X_{3}-\mu_{l_{3}}\right\|^{2}\right.\\ &\left.+\left\|X_{4}-\mu_{l_{1}}\right\|^{2}+\left\|X_{5}-\mu_{l_{2}}\right\|^{2}+\left\|X_{6}-\mu_{l_{1}}\right\|^{2}\right\} \end{aligned}$

You have three  cluster centroids $\left\{\mu_{l_{1}}, \mu_{l_{2}}, \mu_{l_{3}}\right\}=\{(0,1),(4,4),(6,1)\}$. Now you (re-)assign 6 points to one of the 3 clusters with the regard of the centroid. You choose the smallest (squared) distance between the point and the three centroids and assign the corresponding point to the corresponding cluster.

In total you have 6  distances due the 6 points. The other combination/distances are not regarded.
Finally you calculate the arithmetic mean. The squared distance between $X_1$ and the centroid of cluster $3$ is $(6-6)^2+(-1-1)^2=0+4=4$
