[GAP Forum] Computing eigenvalues with multiplicities

[Jean-Claude Arbaut]

To get the multiplicities, you may use

List(Eigenspaces(Rationals, m), Dimension);

Since it uses the output of the function Eigenvalues, the lists should 
have the same order (unless the algorithm is randomized...).

However, it's maybe to much work for GAP to get only the multiplicites.
Then you can check the code for function Eigenvalues in gap4r7\lib\matrix.gi
and adapt it to your needs:

Eigenvalues actually calls GeneralisedEigenvalues, which returns a 
Set(Factors(...)).
Here the factors are

gap> Factors(UnivariatePolynomialRing(Rationals), 
MinimalPolynomial(Rationals, m, 1));
[ x_1-1, x_1+1 ]

If you replace MinimalPolynomial with CharacteristicPolynomial, you get

gap> Factors(UnivariatePolynomialRing(Rationals), 
CharacteristicPolynomial(Rationals, m));
[ x_1-1, x_1-1, x_1+1 ]

Then you can recover the multiplicities.

HTH,

Jean-Claude Arbaut

Le 03/12/2014 16:59, Moritz Schmitt a écrit :
> Dear all,
>
> Let's assume we are given an integer matrix like
>
> m := [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ]
>
> How do we compute all eigenvalues of m with multiplicities? If I use the command
>
> Eigenvalues(Rationals, m);
>
> I only get as output
>
> [1,-1]
>
> but actually need something like [1,1,-1]. I browsed GAP's manual but
> didn't find anything.
>
> Thanks in advance.
>
> Moritz
>
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