Matrix multiplication with submatrices

Matrix multiplication with submatrices

A simple observation:

\begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e \\ f \end{pmatrix} &= \begin{pmatrix} \begin{pmatrix} a & b \end{pmatrix} \begin{pmatrix} e \\ f \end{pmatrix} \\ \begin{pmatrix} c & d \end{pmatrix} \begin{pmatrix} e \\ f \end{pmatrix} \end{pmatrix} \end{align*}

Rewrite it in a more friendly form:

\begin{align*} \begin{pmatrix} w_1^T \\ w_2^T \end{pmatrix} \begin{pmatrix} v \end{pmatrix} &= \begin{pmatrix} w_1^Tv \\ w_2^Tv \\ \end{pmatrix} \end{align*}

Similarly:

\begin{align*} \begin{pmatrix} u^T \end{pmatrix} \begin{pmatrix} w_1 & w_2 \end{pmatrix} &= \begin{pmatrix} u^Tw_1 & u^Tw_2 \end{pmatrix} \end{align*}