Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
frobenius(flag=0, var=’x’)
Return the Frobenius form (rational canonical form) of this matrix.
INPUT:
•flag – 0 (default), 1 or 2 as follows:
–0 – (default) return the Frobenius form of this matrix.
–1 – return only the elementary divisor polynomials, as polynomials in var.
–2 – return a two-components vector [F,B] where F is the Frobenius form and B is the basis change
so that M = B −1 F B.
•var – a string (default: ‘x’)
ALGORITHM: uses PARI’s matfrobenius()
EXAMPLES:
sage: A = MatrixSpace(ZZ, 3)(range(9))
sage: A.frobenius(0)
[ 0 0 0]
[ 1 0 18]
[ 0 1 12]
sage: A.frobenius(1)
[x^3 - 12*x^2 - 18*x]
sage: A.frobenius(1, var=’y’)
[y^3 - 12*y^2 - 18*y]
sage: F, B = A.frobenius(2)
sage: A == B^(-1)*F*B
True
sage: a=matrix([])
sage: a.frobenius(2)
([], [])
sage: a.frobenius(0)
[]
sage: a.frobenius(1)
[]
sage: B = random_matrix(ZZ,2,3)
sage: B.frobenius()
Traceback (most recent call last):
...
ArithmeticError: frobenius matrix of non-square matrix not defined.
AUTHORS:
•Martin Albrect (2006-04-02)
TODO: - move this to work for more general matrices than just over Z. This will require fixing how PARI
polynomials are coerced to Sage polynomials.
gcd()
Return the gcd of all entries of self; very fast.
EXAMPLES:
sage: a = matrix(ZZ,2, [6,15,-6,150])
sage: a.gcd()
3
height()
Return the height of this matrix, i.e., the max absolute value of the entries of the matrix.
325