Manipulate[
VectorPlot[{{α,2},{-2,0}}.{x1,x2},{x1,-3,3},{x2,-3,3},VectorScale->{0.04,1,None},Axes->True],
{{α,0},-6,6,Appearance->"Labeled"}]

Manipulate[StreamPlot[{α*x1+2*x2,-2*x1},{x1,-3,3},{x2,-3,3},VectorScale->{0.04,1,None},Axes->True],{{α,0},-6,6,Appearance->"Labeled"}];

Manipulate[
Grid[
{
{NumberForm[Eigenvalues[(({{α, 2}, {-2, 0}}))],{4,3}]},{StreamPlot[{{α,2},{-2,0}}.{x1,x2},{x1,-3,3},{x2,-3,3},VectorScale->{0.04,1,None},Axes->True]}
}
],
{{α,0},-6,6,Appearance->"Labeled"}]