I was reading Fuzzy clustering with volume prototypes
and adaptive cluster merging by Kaymak, U and Setnes, M.
Here it is written the following equation
\begin{equation}
P_i=\frac{\sum_{k=1}^{N} u_{ik}^m(x_k-v_i)(x_k-v_i)^T}{\sum_{k=1}^{N}u_{ik}^m}
\end{equation}
where i represent the $i^{th}$ cluster
and it is said that above matrix is symmetric.
I don't understand how can i proof this.
You can find a copy of the paper here
Matrix $A$ is said to be symmetric if $\mathbf{A} = \mathbf{A}^T$, where superscript $T$ stands for matrix transpose.
Forget cluster index $i$ (just assume you take one cluster), the formula becomes $$ \mathbf{P} = \frac{\sum_{k=1}^{N}u_{k}^m(\mathbf{x}_k-\mathbf{v})(\mathbf{x}_k-\mathbf{v})^T}{\sum_{k=1}^{N}u_{k}^m}, $$ where $\mathbf{x}_k, \mathbf{v}$ are vectors and $u_{k}^m$ are scalars. Let $\frac{1}{\sum_{k=1}^{N}u_{k}^m} = \beta$ and $\mathbf{a}_k = \sqrt{u_k^m}(\mathbf{x}_k-\mathbf{v})$, then $$ \mathbf{P}=\beta\sum_{k=1}^N \mathbf{a}_k\mathbf{a}_k^T. $$ Hence, $\mathbf{P}$ is symmetric since (using properties of matrix transpose) $$ \mathbf{P}^T = \beta^T\sum_{k=1}^N (\mathbf{a}_k\mathbf{a}_k^T)^T = \beta \sum_{k=1}^N \mathbf{a}_k\mathbf{a}_k^T = \mathbf{P}. $$