Using numerical inaccurate numbers such as 0.3 are not a "proof" that something is wrong in the symbolic comp
Eigenvalues[\[Rho]Gpmatrix[\[Alpha], \[Beta], 3/10]]
returns {0, 0, 7/20, 13/20}
If you check more carefully by
Solve[CharacteristicPolynomial[\[Rho]Gpmatrix[\[Alpha], \[Beta], a],
x] == 0, x]
or
Solve[Det[\[Rho]Gpmatrix[\[Alpha], \[Beta], a] -
x IdentityMatrix[4]] == 0, x]
you will notice that the results only depend on the parameter a
If you want a "numerical proof" just run
Table[Eigenvalues[\[Rho]Gpmatrix[\[Alpha], \[Beta], 0.3]], {\[Alpha],
0, 1, 0.1}, {\[Beta], 0, 1, 0.1}] // Chop
and see that the results do not depend on the values for alpha or beta
yehuda