Here are the eigenvalues for the constants:
{mub, muf, g2, kappas, kappaa, beta, lambda1, lambda2, lambda3, lambda4, lambda5} = 
{0.1803, 0.362, 1, -0.323, 0, 0, 0, 0, 0, 0, 0}

K = 3 and two links with masses g^2, and Kappa_S, nonzero


In[31]:=
expr = N[generalMatrix[makeham, 3, 2, 0, 
        1, {0.1803, 0.362, 1, -0.323, 0, 0, 0, 0, 0, 0, 0}, none]];

In[32]:=
Sort[Chop[Eigenvalues[expr]]]

Out[32]=
{0.417543, 0.417543, 0.417543, 0.417543, 1.49755, 1.49755, 1.49755, 1.49755, \
1.49755, 1.49755, 1.49755, 1.49755, 1.49755, 1.49755, 1.49755, 1.49755, \
1.64288, 1.64288, 1.64288, 1.64288, 1.69473, 1.69473, 1.69473, 1.69473, \
1.78998, 1.78998, 1.78998, 1.78998, 1.78998, 1.78998, 1.78998, 1.78998, \
1.78998, 1.78998, 1.78998, 1.78998, 2.17546, 2.17546, 2.17546, 2.17546, \
2.3012, 2.3012, 2.3012, 2.3012, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, \
2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, \
2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, \
2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, \
2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.57181, 2.70822, \
2.70822, 2.70822, 2.70822, 2.70822, 2.70822, 2.70822, 2.70822, 2.70822, \
2.70822, 2.70822, 2.70822, 2.93413, 2.93413, 2.93413, 2.93413, 2.96657, \
2.96657, 2.96657, 2.96657, 2.96657, 2.96657, 2.96657, 2.96657, 2.96657, \
2.96657, 2.96657, 2.96657, 3.41035, 3.41035, 3.41035, 3.41035, 3.41035, \
3.41035, 3.41035, 3.41035, 3.41035, 3.41035, 3.41035, 3.41035, 3.69949, \
3.69949, 3.69949, 3.69949}

These are the diagonal matrix elements 

In[33]:=
diag = Table[expr[[i, i]], {i, Length[expr]}]

Out[33]=
{1.60012, 1.60012, 1.60012, 1.60012, 1.61194, 1.61194, 1.61194, 1.61194, \
1.60012, 1.60012, 1.60012, 1.60012, 2.14489, 2.14489, 2.14489, 2.14489, \
2.14489, 2.14489, 2.14489, 2.14489, 2.86954, 2.86954, 2.86954, 2.86954, \
2.86954, 2.86954, 2.86954, 2.86954, 2.14489, 2.14489, 2.14489, 2.14489, \
2.14489, 2.14489, 2.14489, 2.14489, 2.86954, 2.86954, 2.86954, 2.86954, \
2.86954, 2.86954, 2.86954, 2.86954, 2.14489, 2.14489, 2.14489, 2.14489, \
2.14489, 2.14489, 2.14489, 2.14489, 2.14489, 2.14489, 2.14489, 2.14489, \
2.14489, 2.14489, 2.14489, 2.14489, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, 2.60502, \
2.60502, 2.60502, 2.60502, 2.60502}

This is a list of the different elements of the matrix

In[35]:=
el = Chop[Union[Flatten[expr]]]

Out[35]=
{-0.499383, -0.477465, -0.119366, 0, 0, 0.00704468, 0.00996269, 0.0166045, \
0.0265672, 0.105549, 0.167928, 0.263874, 1.60012, 1.61194, 2.14489, 2.60502, \
2.86954}