In my research work, I came across the following expression:
$$ \textbf{S}_c = \sum_{k = 1}^K \textbf{H}_k \textbf{p}_c \textbf{u}_k + \textbf{V}_c$$
where dimension of $\textbf{V}_c$ is $M \times L$, dimension of product of $\textbf{H}_k \textbf{p}_c \textbf{u}_k$ is $ M \times L $ and hence dimension of $ \textbf{S}_c$ is $ M \times L $.
Although, I had derived how these dimensions are coming, I am not getting exactly what role summation is performing in above expression.
Any help in this regard will highly appreciated.
On the right-hand side we have a sum of $K+1$ terms, each an $(M\times L)$-matrix. We have \begin{align*} \textbf{S}_c&= \sum_{k = 1}^K \textbf{H}_k \textbf{p}_c \textbf{u}_k + \textbf{V}_c\\ &=\textbf{H}_1 \textbf{p}_c \textbf{u}_1 + \textbf{H}_2 \textbf{p}_c \textbf{u}_2 + \cdots+\textbf{H}_K \textbf{p}_c \textbf{u}_K +\textbf{V}_c \end{align*}