[GAP Forum] Eigensystem for Hermitian matrices

[Jacek M. Holeczek]

Hi,
could you, please, help me with the following problem.

I have a set of 2x2 and 3x3 Hermitian matrices (i.e. matrix == 
complex_conjugate(transpose(matrix))). So, they are all positively 
defined and all their eigenvalues are real positive numbers.

These matrices are always calculated from some 2- and 3-dimensional 
irreducible matrix representations returned by the Repsn 
IrreducibleAffordingRepresentation function (so Cyclotomics are always 
involved?).

It seems that, for a significant amount of cases, I am not able to 
calculate Eigenvalues / Eigenvectors / Eigenspaces in GAP (e.g. I get 
truncated or completely empty lists of eigenvalues). I suspect then 
that, in general, this is simply not possible in GAP (and I will need to 
ask Mathematica to do it afterwards). Could you, please, confirm / deny 
this statement?

Two examples of such matrices are given below (well, the second one is a 
4x4 matrix but I chose it for its "simplicity"; in general I really need 
2x2 and 3x3 cases only):

[ [ 3, 
-E(15)-E(15)^2-E(15)^4-2*E(15)^7-E(15)^8-2*E(15)^11-2*E(15)^13-2*E(15)^14 
], [ 
-2*E(15)-2*E(15)^2-2*E(15)^4-E(15)^7-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14, 
3 ] ]

[ [ 9/2, -3*E(3)-3/2*E(3)^2, 3*E(3)+3/2*E(3)^2, 0 ], [ 
-3/2*E(3)-3*E(3)^2, 9/2, 0, 3/2*E(3)-3/2*E(3)^2 ], [ 3/2*E(3)+3*E(3)^2, 
0, 9/2, 0 ], [ 0, -3/2*E(3)+3/2*E(3)^2, 0, 9/2 ] ]

Thanks in advance,
Best regards,
Jacek.