I have a 6x6 matrix with integer entries, whose eigenvalues are also
integers. I wanted the Jordan canonical form, and the associated
matrix to make the similarity transformation. The Jordan form comes
out nicely, but I can't get the transformation matrix. I've included
the error output below - the error seems more severe without setting
base_ring=QQ. I've also include a legitimate transformation matrix I
worked up by hand (with some help from SAGE!).
Is this expected behavior? Any usage hints or workarounds? Thanks.
Rob
m=matrix(QQ, [[2,0,1,1,0,0],[0,2,1,1,0,0],[2,0,1,0,0,1],[2,0,0,1,1,0],
[0,2,1,0,0,1],[0,2,0,1,1,0]])
m.jordan_form()
[4|0|0 0|0 0]
[-+-+---+---]
[0|2|0 0|0 0]
[-+-+---+---]
[0|0|0 1|0 0]
[0|0|0 0|0 0]
[-+-+---+---]
[0|0|0 0|0 1]
[0|0|0 0|0 0]
p=m.jordan_form(base_ring=QQ, transformation=True)
Traceback (click to the left for traceback)
...
ValueError: cannot compute the basis of the Jordan block of size 2
with
eigenvalue 0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/rob/.sage/sage_notebook/worksheets/admin/46/code/98.py",
line 6, in <module>
p=m.jordan_form(base_ring=QQ, transformation=True)
File "/opt/sage-3.1.2/local/lib/python2.5/site-packages/
SQLAlchemy-0.4.6-py2.5.egg/", line 1, in <module>
File "matrix2.pyx", line 4125, in
sage.matrix.matrix2.Matrix.jordan_form (sage/matrix/matrix2.c:23429)
ValueError: cannot compute the basis of the Jordan block of size 2
with eigenvalue 0
p=matrix(QQ,[[1,1,0,1,3,1],[1,-1,0,1,3,1],[1,0,1,1,0,1],
[1,0,-1,-3,-6,0],[1,-2,1,0,0,-8],[1,-2,-1,-2,-6,-3]])
p.inverse()*m*p == m.jordan_form()
True
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