Find eigenvalues of a matrix?

Questions
,
1

if we defined a matrix us a function of parametrs , the eingenvalues is represented just by one parameter !!!
you can see the file.
thank you in advance

2

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

3

thank you, you are right the eigenvalue is independent of alpha and beta