Usage of elementary matrices and vectors

Usage of elementary matrices and vectors

Given a matrix \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} We want to find a transformation that yields \begin{pmatrix} 0 & 0 & 0 \\ 2a & 2b & 2c \\ 0 & 0 & 0 \\ \end{pmatrix}

Let's do it a small step at a time. First let's single out the first row:

\begin{align*} \begin{pmatrix} a & b & c \\ \end{pmatrix} &= \begin{pmatrix} 1 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} \end{align*}

Then populate this row into the required matrix:

\begin{align*} \begin{pmatrix} 0 & 0 & 0 \\ 2a & 2b & 2c \\ 0 & 0 & 0 \\ \end{pmatrix} &= 2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \begin{pmatrix} a & b & c \\ \end{pmatrix} \end{align*}

Now consider the row operation of adding to a row a multiple of the other row: $a_k := a_k + \alpha a_n$ Building upon the results we have already gotten, it's quite simple:

\begin{align*} & \begin{pmatrix} a & b & c \\ 2a+d & 2b+e & 2c+f \\ g & h & i \\ \end{pmatrix} \\ &= \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0 \\ 2a & 2b & 2c \\ 0 & 0 & 0 \\ \end{pmatrix} \\ &= \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} + 2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} \\ &= (I + 2e_2 e_1^T) \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} \\ \end{align*}