Hello all:
In trying to compute the eigenvalues and eigenvectors of the
discrete Fourier transformation for Z/5Z for a class I'm teaching,
I came across the following:
sage: MS = MatrixSpace(CyclotomicField(5),5,5)
sage: z = CyclotomicField(5).gen()
sage: F5 =
MS([[1,1,1,1,1],[1,z,z^2,z^3,z^4],[1,z^2,z^4,z^6,z^8],[1,z^3,z^6,z^9,z^(12)],[1,z^4,z^8,z^(12),z^(16)]])
sage: F5.eigenspaces()
---------------------------------------------------------------------------
<class 'gen.PariError'> Traceback (most recent call last)
/mnt/hd200/sagefiles/sage-1.5/<ipython console> in <module>()
/mnt/hd200/sagefiles/sage-1.5/matrix2.pyx in matrix2.Matrix.eigenspaces()
/mnt/hd200/sagefiles/sage-1.5/matrix2.pyx in matrix2.Matrix.kernel()
/mnt/hd200/sagefiles/sage-1.5/sage_object.pyx in sage_object.SageObject._pari_()
/mnt/hd200/sagefiles/sage-1.5/gen.pyx in gen._pari_trap()
<class 'gen.PariError'>: forbidden (45)
Is this method only defined over QQ? Note that
sage: A = F5 - (-2*z^3 - 2*z^2 - 1)*MS.identity_matrix()
sage: A.kernel()
works. (BTW, (-2*z^3 - 2*z^2 - 1 = -\sqrt{5}.)
- David Joyner
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