Loading Diagonals

We have been enforcing boundary conditions by loading diagonals.  The plate folks are now using the same techqnique to compute eigenvalues.  The question is what happens to the eigenvalues when we load diagonals.

n=12;
A=RandomReal[{-1,1},{n,n}]+ DiagonalMatrix[RandomReal[{0.5(Sqrt[n]),1.5 (Sqrt[n])},{n}]];
A=A+Transpose[A];
b=RandomReal[{-1,1},n];

Big=1.0 (10^6);
α=1.2;
p=7;
AA=A;bb=b;
AA[[p,p]]=Big;
bb[[p]]=α Big; 
LinearSolve[AA,bb]
LinearSolve[A,b]

ListLogPlot[{Eigenvalues[AA],Eigenvalues[A]},
PlotRange->All]