I understand $AX = X'$ and by doing so, you get both equation for derivative of $x_1 and $x_2$.
When I make a $x_1$ and $x_2$ plot, I am confused regarding which derivative of $x_1$ or $x_2$ to choose?
Say I want to create a direction field for $$X' = \begin{pmatrix} -1/2& 1 \\ -1 & -1/2 \end{pmatrix} X$$ How would I plot a direction field of $x_1$ and $x_2$? $X$ matrix contains $x_1$ and $x_2$.
$\endgroup$ 21 Answer
$\begingroup$You don't choose any derivatives. The direction field consists of vectors $A\vec x$ where $\vec x$ ranges over the plane. For example, at $(2,4)$ you draw the vector $$ \begin{pmatrix} -1/2& 1 \\ -1 & -1/2 \end{pmatrix} \begin{pmatrix} 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ -4 \end{pmatrix} $$
(One usually scales down these vectors; keeping their direction but not the length. Otherwise the plot would be a mess of overlapping arrows.)
You could go on, picking some points with convenient (small integer) coordinates. A more sophisticated approach is to look for nullclines: the lines along with one of two components of $A\vec x$ is zero. Then sketch the field within each of four angles formed by the nullclines.
Or just use a computer, e.g., Desmos vector field generator:
