Figure $1$: Scatter plot of datasets and the initialized centers of $3$ clusters

Given the input matrix $x$ whose rows represent different data points, please perform a k$-$means clustering on this dataset using the Euclidean distance as the distance function. Here $k$ is chosen as $3.$ The Euclidean distance d between input vector vec$(p)in{R}^n$ and vec$(q)in{R}^n$ is defined as

$d=\sqrt[\phantom{2}]{{\displaystyle \underset{i=1}{\overset{n}{}}}(p_i-q_i{)}^2}$

All data in X were plotted in Figure $1.$ The center of $3$ clusters were initialized as vec$(c_1)=(6\mathrm{.}2,3\mathrm{.}2)($red$),$ vec$(c_2)=(6\mathrm{.}6,3\mathrm{.}7)($green$),$ vec$(c_3)=(6\mathrm{.}5,3\mathrm{.}0)($blue$).$

$x=\left[\begin{array}{cc}5\mathrm{.}9& 3\mathrm{.}2\\ 4\mathrm{.}6& 2\mathrm{.}9\\ 6\mathrm{.}2& 2\mathrm{.}8\\ 4\mathrm{.}7& 3\mathrm{.}2\\ 5\mathrm{.}5& 4\mathrm{.}2\\ 5\mathrm{.}0& 3\mathrm{.}0\\ 4\mathrm{.}9& 3\mathrm{.}1\\ 6\mathrm{.}7& 3\mathrm{.}1\\ 5\mathrm{.}1& 3\mathrm{.}8\\ 6\mathrm{.}0& 3\mathrm{.}0\end{array}\right]$

a$.$ What are $3$ clusters and their centers after one iteration? Show the detailed steps, same as questions b$.[15$ pts$]$

b$.$ What are $3$ clusters and their centers after two iterations? $[15$ pts$]$