Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 0 0| 0 0 1 -2| 0 0 0 0 0 0]
[-------+---------------+-----------------------]
[ 0 0| 0 0 0 0| 0 0 0 0 0 50]
[ 0 0| 0 0 0 0| 1 0 0 0 0 40]
[ 0 0| 0 0 0 0| 0 1 0 0 0 3]
[ 0 0| 0 0 0 0| 0 0 1 0 0 -12]
[ 0 0| 0 0 0 0| 0 0 0 1 0 -12]
[ 0 0| 0 0 0 0| 0 0 0 0 1 -4]
sage: F. = QQ[]
sage: polys = A.rational_form(format=’invariants’)
sage: [F(p).factor() for p in polys]
[x^2 + 2*x + 5, (x^2 - 2) * (x^2 + 2*x + 5), (x^2 - 2) * (x^2 + 2*x + 5)^2]
Rational form may be computed over any field. The matrix below is an example where the eigenvalues lie
outside the field.
[[6*a + 5, 1], [6*a + 1, a + 3, a + 3, 1], [5*a, a + 4, a + 6, 6*a + 5, 6*a + 1, 5*a + 6, 5*
NotImplementedError: eigenvalues() is not implemented for matrices with eigenvalues that are
sage: F. = FiniteField(7^2)
sage: A = matrix(F,
... [[5*a + 3, 4*a + 1, 6*a + 2, 2*a + 5, 6, 4*a + 5, 4*a + 5, 5, a + 6,
... [6*a + 3, 2*a + 4, 0, 6, 5*a + 5, 2*a, 5*a + 1, 1, 5*a + 2,
... [3*a + 1, 6*a + 6, a + 6, 2, 0, 3*a + 6, 5*a + 4, 5*a + 6, 5*a + 2,
... [ 3*a, 6*a, 3*a, 4*a, 4*a + 4, 3*a + 6, 6*a, 4, 3*a + 4, 6*a
... [4*a + 5, a + 1, 4*a + 3, 6*a + 5, 5*a + 2, 5*a + 2, 6*a, 4*a + 6, 6*a + 4, 5*a
... [ 3*a, 6*a, 4*a + 1, 6*a + 2, 2*a + 5, 4*a + 6, 2, a + 5, 2*a + 4, 2*a
... [4*a + 5, 3*a + 3, 6, 4*a + 1, 4*a + 3, 6*a + 3, 6, 3*a + 3, 3, a
... [6*a + 6, a + 4, 2*a + 6, 3*a + 5, 4*a + 3, 2, a, 3*a + 4, 5*a, 2*a
... [3*a + 5, 6*a + 2, 4*a, a + 5, 0, 5*a, 6*a + 5, 2*a + 1, 3*a + 1, 3*a
... [3*a + 2, a + 3, 3*a + 6, a, 3*a + 5, 5*a + 1, 3*a + 2, a + 3, a + 2, 6*a
... [6*a + 6, 5*a + 1, 4*a, 2, 5*a + 5, 3*a + 5, 3*a + 1, 2*a, 2*a, 2*a
sage: A.rational_form()
[ a + 2| 0 0 0| 0 0 0 0 0 0 0]
[-------+-----------------------+-------------------------------------------------------]
[ 0| 0 0 a + 6| 0 0 0 0 0 0 0]
[ 0| 1 0 6*a + 4| 0 0 0 0 0 0 0]
[ 0| 0 1 6*a + 4| 0 0 0 0 0 0 0]
[-------+-----------------------+-------------------------------------------------------]
[ 0| 0 0 0| 0 0 0 0 0 0 2*a]
[ 0| 0 0 0| 1 0 0 0 0 0 6*a + 3]
[ 0| 0 0 0| 0 1 0 0 0 0 6*a + 1]
[ 0| 0 0 0| 0 0 1 0 0 0 a + 2]
[ 0| 0 0 0| 0 0 0 1 0 0 a + 6]
[ 0| 0 0 0| 0 0 0 0 1 0 2*a + 1]
[ 0| 0 0 0| 0 0 0 0 0 1 2*a + 1]
sage: invariants = A.rational_form(format=’invariants’)
sage: invariants
sage: R. = F[]
sage: polys = [R(p) for p in invariants]
sage: [p.factor() for p in polys]
[x + 6*a + 5, (x + 6*a + 5) * (x^2 + (2*a + 5)*x + 5*a), (x + 6*a + 5) * (x^2 + (2*a + 5)*x
sage: polys[-1] == A.minimal_polynomial()
True
sage: prod(polys) == A.characteristic_polynomial()
True
sage: A.eigenvalues()
Traceback (most recent call last):
...
228 Chapter 7. Base class for matrices, part 2