(* The case where β=0 *)
(* There is no need to manipulate the ω parameter because we can simply measure time in units such that ω=1 *)	
Manipulate[ParametricPlot[{{Cos[t],-Sin[t]},{Sin[t],Cos[t]}}.{A,B},{t,0,2π},PlotRange->{-2,2}],{A,1,0},{B,1,0}]

(* The case where α is not equal to β: *)

Manipulate[ParametricPlot[Exp[- β t](Cos[Sqrt[(α^2)-(β^2)] t]IdentityMatrix[2]+(1/(Sqrt[(α^2)-(β^2)]))Sin[Sqrt[(α^2)-(β^2)]t]{{β,α},{-α,-β}}).{A,B},{t,0,2π},PlotRange->{-2,2}],{A,1,0},{B,1,0},{{α,8.14},10,0},{{β,0.48},10,0}]

(* The case where α is equal to β *)

Manipulate[ParametricPlot[Exp[- β t](IdentityMatrix[2]+t{{β,β},{-β,-β}}).{A,B},{t,0,2π},PlotRange->{-2,2}],{A,1,0},{B,1,0},{{β,0.48},100,0}]