Matrix determinant and simplify

[proy87]

I want to calculate eigenvalues "manually", so I form the following matrix:
m=Matrix([[Rational(1,2)-x, sqrt(3)/2, 0,0], [1, sqrt(3)-x,0,0],[-sqrt(3)/2, Rational(1,2), -x, 0],[-2,0,0,-x]])
m.det() gives (4*x**5 - 8*sqrt(3)*x**4 - 4*x**4 + 4*sqrt(3)*x**3 + 13*x**3)/(4*x - 4*sqrt(3) - 2), when, in fact, it should be a polynomial and I can't find a way to simplify it: factor and cancel don't work.

[oscarbenjamin]

You can use extension=True:

In [3]: m.det()
Out[3]: 
   5         4      4         3       3
4⋅x  - 8⋅√3⋅x  - 4⋅x  + 4⋅√3⋅x  + 13⋅x 
───────────────────────────────────────
             4⋅x - 4⋅√3 - 2            

In [4]: factor(m.det(), extension=True)
Out[4]: 
 3               
x ⋅(x - √3 - 1/2)

In [5]: cancel(m.det(), extension=True)
Out[5]: 
      3            
 4   x ⋅(-4⋅√3 - 2)
x  + ──────────────
           4   

The algorithm for charpoly is different and does not use division:

In [6]: m.charpoly()
Out[6]: 
PurePoly(lambda**4 + (4*x - sqrt(3) - 1/2)*lambda**3 + 3*x*(4*x - 2*sqrt(3) - 1)/2*lambda**2 + x**2*(8*x - 6*sqrt(3) - 3)/2*lamb
da + x**3*(x - sqrt(3) - 1/2), lambda, domain='EX')

[proy87]

@oscarbenjamin , I use det(method='berkowitz') which works. Can it give division? Is there any reason that the default algorithm for det is bareiss, not berkowitz?

[oscarbenjamin]

The bareiss algorithm is supposed to use exact division i.e. divisions that always cancel. I guess that it isn't using cancel or isn't using extension=True so the cancellation doesn't happen. It would make more sense to use the bareiss algorithm with DomainMatrix.