A= SparseArray[ Band[{2,1}]->RandomReal[],{12,12}]
B= SparseArray[Band[{3,2}]->RandomReal[{-1,1}],{12,12}]
CC=A+B
CC[[2,3]]+=2.1
CC[[1,1]]==1.3

DD=Normal[CC]
SparseArray[DD]

OriginalToFake[γ_]:=(({{10000.`, 0, 0, -2.`, 0, -1.`, 0, 0, 0, 0, 0, 0, -2.`, 0}, {0, 10000.`, 0, -2.`, -1.`, 0, 0, -2.`, 0, 0, 0, 0, 0, 0}, {0, 0, 4.`, 0, -1.`, -1.`, 0, 0, -1.`, 0, 0, -1.`, 0, 0}, {-2.`, -2.`, 0, 10000.`, -1.`, -1.`, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -1.`, -1.`, -1.`, 12.`, -4.`, 0, -1.`, -4.`, 0, 0, 0, 0, 0}, {-1.`, 0, -1.`, -1.`, -4.`, 12.`, 0, 0, 0, 0, 0, -4.`, -1.`, 0}, {0, 0, 0, 0, 0, 0, 10000.`, -2.`, -1.`, 0, -2.`, 0, 0, 0}, {0, -2.`, 0, 0, -1.`, 0, -2.`, 10000.`, -1.`, 0, 0, 0, 0, 0}, {0, 0, -1.`, 0, -4.`, 0, -1.`, -1.`, 12.`, 0, -1.`, -4.`, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 10000.`, -2.`, -1.`, -2.`, 0}, {0, 0, 0, 0, 0, 0, -2.`, 0, -1.`, -2.`, 10000.`, -1.`, 0, 0}, {0, 0, -1.`, 0, 0, -4.`, 0, 0, -4.`, -1.`, -1.`, 12.`, -1.`, 0}, {-2.`, 0, 0, 0, 0, -1.`, 0, 0, 0, -2.`, 0, -1.`, 10000.`, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}))+N[γ] (({{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, (1/2), 0, 0, 0, 0, ((-1)/2)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (1/2), 0, ((-1)/2)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, ((-1)/2), 0, 0, ((-1)/2), 0, 1}}));
FakeToReal[γ_]:=OriginalToFake[1]+N[γ] ((({{10000.`, 0, 0, -2.`, 0, -1.`, 0, 0, 0, 0, 0, 0, -2.`, 0}, {0, 10000.`, 0, -2.`, -1.`, 0, 0, -2.`, 0, 0, 0, 0, 0, 0}, {0, 0, 8.`, 0, -1.`, -1.`, 0, 0, -2.5`, 0, 0, -2.5`, 0, -1}, {-2.`, -2.`, 0, 10000.`, -1.`, -1.`, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -1.`, -1.`, -1.`, 12.`, -4.`, 0, -1.`, -4.`, 0, 0, 0, 0, 0}, {-1.`, 0, -1.`, -1.`, -4.`, 12.`, 0, 0, 0, 0, 0, -4.`, -1.`, 0}, {0, 0, 0, 0, 0, 0, 10000.`, -2.`, -1.`, 0, -2.`, 0, 0, 0}, {0, -2.`, 0, 0, -1.`, 0, -2.`, 10000.`, -1.`, 0, 0, 0, 0, 0}, {0, 0, -2.5`, 0, -4.`, 0, -1.`, -1.`, 12.`, 0, -2.5`, 0.`, 0, -1.`}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 10000.`, -2.`, -1.`, -2.`, 0}, {0, 0, 0, 0, 0, 0, -2.`, 0, -2.5`, -2.`, 10000.`, -2.5`, 0, -1}, {0, 0, -2.5`, 0, 0, -4.`, 0, 0, 0.`, -1.`, -2.5`, 12.`, -1.`, -1.`}, {-2.`, 0, 0, 0, 0, -1.`, 0, 0, 0, -2.`, 0, -1.`, 10000.`, 0}, {0, 0, -1, 0, 0, 0, 0, 0, -1.`, 0, -1, -1.`, 0, 4.`}}))-OriginalToFake[1]);

Evals[γ_] :=Drop[Eigenvalues[If[γ<1,OriginalToFake[γ],FakeToReal[γ-1]]],8]
LogPlot[ Evals[γ],{γ,0,2}]

Evecs[γ_]:= Chop[Eigenvectors[If[γ<1,OriginalToFake[γ],FakeToReal[γ-1]]],(10^(-6))];

Drop

Normalize

Manipulate[
MatrixPlot[
Thing=Evecs[γ][[{10,11,12,13,14}]];
Thing=Map[Drop[#,8]&,Thing];
Thing=Map[ Normalize, Thing];
Thing=Sign[Thing[[All,1]]] Thing;
Thing
],{γ,1,2,0.1}]

Manipulate[Eigenvalues[If[γ<1,OriginalToFake[γ],FakeToReal[γ-1]]]//MatrixForm,{γ,0,2,0.1}]
Manipulate[Chop[Eigenvectors[If[γ<1,OriginalToFake[γ],FakeToReal[γ-1]]],(10^(-6))]//MatrixForm,{γ,0,2,0.1}]

Manipulate[Chop[Eigenvectors[If[γ<1,OriginalToFake[γ],FakeToReal[γ-1]]],(10^(-6))]//MatrixPlot,{γ,1,2,0.1}]

ListSurfacePlot3D