Answer by Carl Woll for Turn vector equation into matrix multiplication

You could use CoefficientArrays. For example:

m = CoefficientArrays[    {    Ay[i]-Ay[i+1],    -Ax[i]+Ax[i+1]-a Az[i],    b Ay[i]    },    {Ax[i],Ay[i],Az[i],Ax[i+1],Ay[i+1],Az[i+1]}][[2]];m //Normal //TeXForm

$\left(\begin{array}{cccccc} 0 & 1 & 0 & 0 & -1 & 0 \\ -1 & 0 & -a & 1 & 0 & 0 \\ 0 & b & 0 & 0 & 0 & 0 \\\end{array}\right)$