Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: A = Mat(GF(7),3,3)(range(3)*3)
sage: A.charpoly()
x^3 + 4*x^2
sage: A = Mat(Integers(6),3,3)(range(9))
sage: A.charpoly()
x^3
determinant()
Return the determinant of this matrix.
EXAMPLES:
sage: m = matrix(GF(101),5,range(25))
sage: m.det()
0
sage: m = matrix(Integers(4), 2, [2,2,2,2])
sage: m.det()
0
TESTS:
sage: m = random_matrix(GF(3), 3, 4)
sage: m.determinant()
Traceback (most recent call last):
...
ValueError: self must be a square matrix
echelonize(algorithm=’gauss’, **kwds)
Puts self in row echelon form.
INPUT:
•self - a mutable matrix
•algorithm- ’gauss’ - uses a custom slower O(n 3 ) Gauss elimination implemented in Sage.
•**kwds - these are all ignored
OUTPUT:
•self is put in reduced row echelon form.
•the rank of self is computed and cached
•the pivot columns of self are computed and cached.
•the fact that self is now in echelon form is recorded and cached so future calls to echelonize return
immediately.
EXAMPLES:
sage: a = matrix(GF(97),3,4,range(12))
sage: a.echelonize(); a
[ 1 0 96 95]
[ 0 1 2 3]
[ 0 0 0 0]
sage: a.pivots()
(0, 1)
hessenbergize()
Transforms self in place to its Hessenberg form.
295