Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: print decomp
2*B[[1, 4, 2, 3, 5]] + 3*B[[3, 1, 4, 2, 5]] + 9*B[[4, 1, 3, 5, 2]] + 6*B[[5, 3, 4, 1, 2]]
An exception is raised when the matrix is not bistochastic:
sage: M = Matrix([[2,3],[2,2]])
sage: decomp = sage.combinat.permutation.bistochastic_as_sum_of_permutations(M)
Traceback (most recent call last):
...
ValueError: The matrix is not bistochastic
characteristic_polynomial(*args, **kwds)
Synonym for self.charpoly(...).
EXAMPLES:
sage: a = matrix(QQ, 2,2, [1,2,3,4]); a
[1 2]
[3 4]
sage: a.characteristic_polynomial(’T’)
T^2 - 5*T - 2
charpoly(var=’x’, algorithm=None)
Returns the characteristic polynomial of self, as a polynomial over the base ring.
ALGORITHM:
In the generic case of matrices over a ring (commutative and with unity), there is a division-free algorithm,
which can be accessed using "df", with complexity O(n 4 ). Alternatively, by specifying
"hessenberg", this method computes the Hessenberg form of the matrix and then reads off the characteristic
polynomial. Moreover, for matrices over number fields, this method can use PARI’s charpoly
implementation instead.
The method’s logic is as follows: If no algorithm is specified, first check if the base ring is a number
field (and then use PARI), otherwise check if the base ring is the ring of integers modulo n (in which
case compute the characteristic polynomial of a lift of the matrix to the integers, and then coerce back
to the base), next check if the base ring is an exact field (and then use the Hessenberg form), or otherwise,
use the generic division-free algorithm. If an algorithm is specified explicitly, if algorithm ==
"hessenberg", use the Hessenberg form, or otherwise use the generic division-free algorithm.
The result is cached.
INPUT:
•var - a variable name (default: ‘x’)
•algorithm - string:
EXAMPLES:
– "df" - Generic O(n 4 ) division-free algorithm
– "hessenberg" - Use the Hessenberg form of the matrix
First a matrix over Z:
sage: A = MatrixSpace(ZZ,2)( [1,2, 3,4] )
sage: f = A.charpoly(’x’)
sage: f
x^2 - 5*x - 2
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: f(A)
132 Chapter 7. Base class for matrices, part 2