Clear[v11,voff,v22]

tmat={{1,0},{0,1}}

vmat={{v11,voff},{voff,v22}}

MatrixForm[vmat]

det[lam_]:=Det[vmat-lam*tmat]

det[lambda]

la=Solve[det[lamda]==0,lamda]

lambda1[v11_,v22_,voff_]:=(1/2) (v11+v22-(Sqrt[(v11^2)-2 v11 v22+(v22^2)+4 (voff^2)]))

lambda2[v11_,v22_,voff_]:=(1/2) (v11+v22+(Sqrt[(v11^2)-2 v11 v22+(v22^2)+4 (voff^2)]))

a11[v11_,v22_,voff_]:=Sqrt[voff^2/(voff^2+(1/2*((v22-v11)+Sqrt[(v22-v11)^2+4*voff^2]))^2)]

a21[v11_,v22_,voff_]:=Sqrt[voff^2/(voff^2+(1/2*((v22-v11)-Sqrt[(v22-v11)^2+4*voff^2]))^2)]

a12[v11_,v22_,voff_]:=1/2*((v22-v11)+Sqrt[(v22-v11)^2+4*voff^2])/Sqrt[(voff^2+(1/2*((v22-v11)+Sqrt[(v22-v11)^2+4*voff^2]))^2)]

a22[v11_,v22_,voff_]:=1/2*((v22-v11)-Sqrt[(v22-v11)^2+4*voff^2])/Sqrt[(voff^2+(1/2*((v22-v11)-Sqrt[(v22-v11)^2+4*voff^2]))^2)]

{Slider[Dynamic[v11],{0,1,0.1}],Style[NumberForm[Dynamic[v11],{3,2}]]}

{Slider[Dynamic[v22],{0,1,0.1}],Style[NumberForm[Dynamic[v22],{3,2}]]}

Dynamic[Plot[{lambda1[v11,v22,voff],lambda2[v11,v22,voff]},{voff,-2,2}]]

Dynamic[Plot[{a11[v11,v22,voff],a12[v11,v22,voff]},{voff,-3,3},PlotRange->{-1,1}]]

Dynamic[Plot[{a21[v11,v22,voff],a22[v11,v22,voff]},{voff,-3,3},PlotRange->{-1,1}]]