Important Mathematical Operations
$$
\begin{equation}
\mathbf{y}=\mathbf{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j,
\tag{3}
\end{equation}
$$
matrix-matrix multiplication
$$
\begin{equation}
\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj},
\tag{4}
\end{equation}
$$
and transposition
$$
\begin{equation}
\mathbf{A}=\mathbf{B}^T \Longrightarrow a_{ij} = b_{ji}
\tag{5}
\end{equation}
$$