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, 0.162, 0, 0, 0, 0, 0, 0}

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

In[139]:=
expr = N[generalMatrix[makeham, 3, 2, 0, 
        1, {0.1803, 0.362, 1, -0.323, 0.162, 0, 0, 0, 0, 0, 0}, none]];

In[141]:=
Sort[Chop[Eigenvalues[expr]]]

Out[141]=
{0.401808, 0.412538, 0.412538, 0.423849, 1.47764, 1.48437, 1.48437, 1.49535, \
1.49535, 1.49535, 1.49535, 1.49535, 1.49535, 1.51972, 1.51972, 1.54327, \
1.64569, 1.67489, 1.67776, 1.67776, 1.68259, 1.68259, 1.69136, 1.70785, \
1.7811, 1.7811, 1.7811, 1.7811, 1.7811, 1.7811, 1.79375, 1.79375, 1.80491, \
1.80518, 1.80518, 1.82767, 2.18563, 2.19146, 2.19146, 2.19799, 2.28166, \
2.29094, 2.29094, 2.30207, 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, 2.59568, 2.59568, 2.59568, 2.59568, 2.59568, \
2.59568, 2.5991, 2.5991, 2.60076, 2.60076, 2.60456, 2.60456, 2.60456, \
2.60456, 2.60456, 2.60456, 2.61951, 2.62868, 2.62868, 2.63669, 2.70995, \
2.70995, 2.70995, 2.70995, 2.70995, 2.70995, 2.7264, 2.7264, 2.73569, \
2.73569, 2.74196, 2.76202, 2.92865, 2.92963, 2.92963, 2.9331, 2.97611, \
2.97611, 2.97611, 2.97611, 2.97611, 2.97611, 2.99726, 2.99726, 3.00148, \
3.00148, 3.01709, 3.02514, 3.42035, 3.42035, 3.42035, 3.42035, 3.42035, \
3.42035, 3.42689, 3.42689, 3.43121, 3.43121, 3.43294, 3.44118, 3.71631, \
3.71822, 3.71822, 3.72022}

These are the diagonal matrix elements 

In[142]:=
diag = Table[expr[[i, i]], {i, Length[expr]}]

Out[142]=
{1.61516, 1.61516, 1.61516, 1.61516, 1.64536, 1.64536, 1.64536, 1.64536, \
1.61516, 1.61516, 1.61516, 1.61516, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.87204, 2.87204, 2.87204, 2.87204, \
2.87204, 2.87204, 2.87204, 2.87204, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.87204, 2.87204, 2.87204, 2.87204, \
2.87204, 2.87204, 2.87204, 2.87204, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, 2.16828, \
2.16828, 2.16828, 2.16828, 2.16828, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, 2.61337, \
2.61337, 2.61337, 2.61337, 2.61337}

This is a list of the different elements of the matrix

In[140]:=
Union[Flatten[expr]]

Out[140]=
{-0.4976111526621536, -0.477464829275686, -0.1193662073189215, -0.06617266787514924, 
  -0.05614937062046789, -0.016655883104453036, -0.01323453357502985, -0.00999352986267182, 
  -0.008327941552226518 - 0.008327941552226518*I, -0.008327941552226518 + 
   0.008327941552226518*I, -0.007066492733885511, -0.00499676493133591, 
  -0.00499676493133591 - 0.00499676493133591*I, -0.00499676493133591 + 0.00499676493133591*I, 
  -0.0035332463669427556 - 0.0035332463669427556*I, -0.0035332463669427556 + 
   0.0035332463669427556*I, -0.002498382465667955 - 0.002498382465667955*I, 
  -0.002498382465667955 + 0.002498382465667955*I, 0., 0. - 0.00499676493133591*I, 
  0. + 0.00499676493133591*I, 0. - 0.007066492733885511*I, 0. + 0.007066492733885511*I, 
  0. - 0.00999352986267182*I, 0. + 0.00999352986267182*I, 0. - 0.01323453357502985*I, 
  0. + 0.01323453357502985*I, 0. - 0.016655883104453036*I, 0. + 0.016655883104453036*I, 
  0. - 0.05614937062046789*I, 0. + 0.05614937062046789*I, 0. - 0.06617266787514924*I, 
  0. + 0.06617266787514924*I, 0.002498382465667955 - 0.002498382465667955*I, 
  0.002498382465667955 + 0.002498382465667955*I, 0.0035332463669427556 - 
   0.0035332463669427556*I, 0.0035332463669427556 + 0.0035332463669427556*I, 
  0.00499676493133591, 0.00499676493133591 - 0.00499676493133591*I, 
  0.00499676493133591 + 0.00499676493133591*I, 0.005272589966034574, 
  0.007044682571126605 - 0.0017720926050920319*I, 0.007044682571126605 + 
   0.0017720926050920319*I, 0.007066492733885511, 0.007456568238798389, 
  0.008327941552226518 - 0.008327941552226518*I, 0.008327941552226518 + 
   0.008327941552226518*I, 0.00996268563470061 + 0.*I, 
  0.00996268563470061 - 0.0025061173959022207*I, 0.00996268563470061 + 
   0.0025061173959022207*I, 0.00999352986267182, 0.01242761373133065, 0.01323453357502985, 
  0.01660447605783435 - 0.004176862326503702*I, 0.01660447605783435 + 0.004176862326503702*I, 
  0.016655883104453036, 0.01988418197012904, 0.02078133838433805, 
  0.02656716169253496 - 0.0016707449306014807*I, 0.02656716169253496 + 
   0.0016707449306014807*I, 0.02656716169253496 - 0.006682979722405923*I, 
  0.02656716169253496 + 0.006682979722405923*I, 0.05614937062046789, 0.06617266787514924, 
  0.10554948999344792, 0.16792821028158453, 0.26387372498361983, 1.6151606547782247, 
  1.6453563656260766, 2.168277470950887, 2.61337030336696, 2.8720431068561347}