Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: all(polys[i].divides(polys[i+1]) for i in range(len(polys)-1))
True
sage: polys[-1] == A.minimal_polynomial(var=’x’)
True
sage: prod(polys) == A.characteristic_polynomial(var=’x’)
True
Rational form is a canonical form. Any two matrices are similar if and only if their rational forms are
equal. By starting with Jordan canonical forms, the matrices C and D below were built as similar matrices,
while E was built to be just slightly different. All three matrices have equal characteristic polynomials
though E‘s minimal polynomial differs.
sage: C = matrix(QQ, [[2, 31, -10, -9, -125, 13, 62, -12],
... [0, 48, -16, -16, -188, 20, 92, -16],
... [0, 9, -1, 2, -33, 5, 18, 0],
... [0, 15, -5, 0, -59, 7, 30, -4],
... [0, -21, 7, 2, 84, -10, -42, 5],
... [0, -42, 14, 8, 167, -17, -84, 13],
... [0, -50, 17, 10, 199, -23, -98, 14],
... [0, 15, -5, -2, -59, 7, 30, -2]])
sage: C.minimal_polynomial().factor()
(x - 2)^2
sage: C.characteristic_polynomial().factor()
(x - 2)^8
sage: C.rational_form()
[ 0 -4| 0 0| 0 0| 0 0]
[ 1 4| 0 0| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 -4| 0 0| 0 0]
[ 0 0| 1 4| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 0| 0 -4| 0 0]
[ 0 0| 0 0| 1 4| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 0| 0 0| 0 -4]
[ 0 0| 0 0| 0 0| 1 4]
sage: D = matrix(QQ, [[ -4, 3, 7, 2, -4, 5, 7, -3],
... [ -6, 5, 7, 2, -4, 5, 7, -3],
... [ 21, -12, 89, 25, 8, 27, 98, -95],
... [ -9, 5, -44, -11, -3, -13, -48, 47],
... [ 23, -13, 74, 21, 12, 22, 85, -84],
... [ 31, -18, 135, 38, 12, 47, 155, -147],
... [-33, 19, -138, -39, -13, -45, -156, 151],
... [ -7, 4, -29, -8, -3, -10, -34, 34]])
sage: D.minimal_polynomial().factor()
(x - 2)^2
sage: D.characteristic_polynomial().factor()
(x - 2)^8
sage: D.rational_form()
[ 0 -4| 0 0| 0 0| 0 0]
[ 1 4| 0 0| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 -4| 0 0| 0 0]
[ 0 0| 1 4| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 0| 0 -4| 0 0]
[ 0 0| 0 0| 1 4| 0 0]
226 Chapter 7. Base class for matrices, part 2