strange eigenvalue/eigenvector output
John Cremona
ask for T.eigenvalues() I get a list of 6 elements of QQbar, and
T.eigenmatrix_right() returns a pair of 6x6 matrices.
Please tell me that this is a bug and not a feature!
John
sage: TK.parent()
Full MatrixSpace of 3 by 3 dense matrices over Number Field in a with
defining polynomial x^2 + 7
sage: TK
[ 1 -a 2]
[ -1 1/2*a - 1/2 -1]
[ -1 1/2*a - 3/2 -1]
sage: len(TK.eigenvalues())
6
sage: TK.eigenmatrix_left()[0].parent()
Full MatrixSpace of 6 by 6 sparse matrices over Algebraic Field
Jason Grout
Further, look at the dimension of the eigenvector matrix:
sage: R=QuadraticField(-7,'a')
sage: a=R.0
sage: TK=matrix(3,[1,-a,2,-1,a/2-1/2,-1,-1,a/2-3/2,-1])
sage: D,P=TK.eigenmatrix_left()
sage: D.parent()
Full MatrixSpace of 6 by 6 sparse matrices over Algebraic Field
sage: P.parent()
Full MatrixSpace of 6 by 3 dense matrices over Algebraic Field
Jason
John Cremona
It takes the chapr poly, factors it, and for any factor of degree > 1
it constructs the root-field and then finds all the Galois conjugates
of that in QQbar. The problem is that that gives all the conjugates
over QQ, not just the ones over the field of definition.
A ticket will be created...but I can see a problem if the base field
does not have an emebdding into QQbar (i.e. into CC) already, since
then the eigenvalues are not well-defined as elements of QQbar. This
will create ambiguities whenever the base field is not Galois.
John
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