[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.

[Hulpke,Alexander]

Dear Forum, Dear Jacek Holczek,

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?).

Typically representations are constructed by algorithms by inducing representations up from subgroups with easier structure. These representations of subgroups are often linear and then involve cyclotomic numbers, this carries over to the induced representations.

In many cases these cyclotomic numbers are (at least to some extent) an artifect of the method and do not imply that the representation could not be written over a subfield after a suitable base change.

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).

You might be thinking of working over the real or complex numbers. GAP does not know these fields, but by default calculates eigenvalues over the field of matrix coefficients. This does not need to contain eigenvalues.

By specifying a larger field, that is known to contain eigenvalues, GAP will compute these.

Take your first example.

[ [ 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 ] ]

We calculate the characteristic polynomial of this matrix as

gap> c:=CharacteristicPolynomial(m);
x_1^2-6*x_1+3
gap> Discriminant(c);
24

so we know that the polynomial splits over CF(24). Indeed (using that Lcm(15,24)=120) we get:

gap> Eigenvalues(CF(120),m);
[ E(24)-3*E(24)^8-E(24)^11-3*E(24)^16-E(24)^17+E(24)^19,
-E(24)-3*E(24)^8+E(24)^11-3*E(24)^16+E(24)^17-E(24)^19 ]

Caveat 1: Unless the matrices have finite order, there is no guarantee that the matrices will diagonalize over a cyclotomic field.

Caveat 2: GAP cannot factor polynomials over the abstract `Cyclotomics`, so `Eigenvalues(Cyclotomics,m);` will not work but you need to specify a field explicitly.

Caveat 3: While primes dividing the discriminant are a good bet for finding a splitting field, in degrees >2 I am not aware of a nice method that would guarantee the cyclotomic splitting field (that’s the reason for Caveat 2). So some trial might be needed.

Regards,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: ***@colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke