Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
These coefficients include the leading one of the monic polynomial and are ready to be coerced into any
polynomial ring over the same field (see examples of this below). This return value is intended to be the
most compact representation and the easiest to use for testing equality of rational forms.
Because the minimal and characteristic polynomials of a companion matrix are the associated polynomial,
it is easy to see that the product of the polynomials of the blocks will be the characteristic polynomial and
the final polynomial will be the minimal polynomial of the entire matrix.
ALGORITHM:
We begin with ZigZag form, which is due to Arne Storjohann and is documented at zigzag_form().
Then we eliminate ‘’corner” entries enroute to rational form via an additional algorithm of Storjohann’s
[STORJOHANN-EMAIL].
EXAMPLES:
The lists of coefficients returned with the invariants keyword are designed to easily convert to the
polynomials associated with the companion matrices. This is illustrated by the construction below of the
polys list. Then we can test the divisibility condition on the list of polynomials. Also the minimal and
characteristic polynomials are easy to determine from this list.
sage: A = matrix(QQ, [[ 11, 14, -15, -4, -38, -29, 1, 23, 14, -63, 17, 24, 36, 32],
... [ 18, 6, -17, -11, -31, -43, 12, 26, 0, -69, 11, 13, 17, 24],
... [ 11, 16, -22, -8, -48, -34, 0, 31, 16, -82, 26, 31, 39, 37],
... [ -8, -18, 22, 10, 46, 33, 3, -27, -12, 70, -19, -20, -42, -31],
... [-13, -21, 16, 10, 52, 43, 4, -28, -25, 89, -37, -20, -53, -62],
... [ -2, -6, 0, 0, 6, 10, 1, 1, -7, 14, -11, -3, -10, -18],
... [ -9, -19, -3, 4, 23, 30, 8, -3, -27, 55, -40, -5, -40, -69],
... [ 4, -8, -1, -1, 5, -4, 9, 5, -11, 4, -14, -2, -13, -17],
... [ 1, -2, 16, -1, 19, -2, -1, -17, 2, 19, 5, -25, -7, 14],
... [ 7, 7, -13, -4, -26, -21, 3, 18, 5, -40, 7, 15, 20, 14],
... [ -6, -7, -12, 4, -1, 18, 3, 8, -11, 15, -18, 17, -15, -41],
... [ 5, 11, -11, -3, -26, -19, -1, 14, 10, -42, 14, 17, 25, 23],
... [-16, -15, 3, 10, 29, 45, -1, -13, -19, 71, -35, -2, -35, -65],
... [ 4, 2, 3, -2, -2, -10, 1, 0, 3, -11, 6, -4, 6, 17]]
sage: A.rational_form()
[ 0 -4| 0 0 0 0| 0 0 0 0 0 0 0 0]
[ 1 4| 0 0 0 0| 0 0 0 0 0 0 0 0]
[---------+-------------------+---------------------------------------]
[ 0 0| 0 0 0 12| 0 0 0 0 0 0 0 0]
[ 0 0| 1 0 0 -4| 0 0 0 0 0 0 0 0]
[ 0 0| 0 1 0 -9| 0 0 0 0 0 0 0 0]
[ 0 0| 0 0 1 6| 0 0 0 0 0 0 0 0]
[---------+-------------------+---------------------------------------]
[ 0 0| 0 0 0 0| 0 0 0 0 0 0 0 -216]
[ 0 0| 0 0 0 0| 1 0 0 0 0 0 0 108]
[ 0 0| 0 0 0 0| 0 1 0 0 0 0 0 306]
[ 0 0| 0 0 0 0| 0 0 1 0 0 0 0 -271]
[ 0 0| 0 0 0 0| 0 0 0 1 0 0 0 -41]
[ 0 0| 0 0 0 0| 0 0 0 0 1 0 0 134]
[ 0 0| 0 0 0 0| 0 0 0 0 0 1 0 -64]
[ 0 0| 0 0 0 0| 0 0 0 0 0 0 1 13]
sage: R = PolynomialRing(QQ, ’x’)
sage: invariants = A.rational_form(format=’invariants’)
sage: invariants
[[4, -4, 1], [-12, 4, 9, -6, 1], [216, -108, -306, 271, 41, -134, 64, -13, 1]]
sage: polys = [R(p) for p in invariants]
sage: [p.factor() for p in polys]
[(x - 2)^2, (x - 3) * (x + 1) * (x - 2)^2, (x + 1)^2 * (x - 3)^3 * (x - 2)^3]
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