Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: a = matrix(QQ, 4, range(16)); a[0,0] = 1/19; a[0,1] = 1/5
sage: a.echelonize(algorithm=’multimodular’); a
[ 1 0 0 -76/157]
[ 0 1 0 -5/157]
[ 0 0 1 238/157]
[ 0 0 0 0]
height()
Return the height of this matrix, which is the maximum of the absolute values of all numerators and
denominators of entries in this matrix.
OUTPUT: an Integer
EXAMPLES:
sage: b = matrix(QQ,2,range(6)); b[0,0]=-5007/293; b
[-5007/293 1 2]
[ 3 4 5]
sage: b.height()
5007
invert(check_invertible=True, algorithm=’default’)
Compute the inverse of this matrix.
Warning: This function is deprecated. Use inverse instead.
EXAMPLES:
sage: a = matrix(QQ,3,range(9))
sage: a.invert()
Traceback (most recent call last):
...
ZeroDivisionError: input matrix must be nonsingular
minpoly(var=’x’, algorithm=’linbox’)
Return the minimal polynomial of this matrix.
INPUT:
•var - ‘x’ (string)
•algorithm - ‘linbox’ (default) or ‘generic’
OUTPUT: a polynomial over the rational numbers.
EXAMPLES:
sage: a = matrix(QQ, 3, [4/3, 2/5, 1/5, 4, -3/2, 0, 0, -2/3, 3/4])
sage: f = a.minpoly(); f
x^3 - 7/12*x^2 - 149/40*x + 97/30
sage: a = Mat(ZZ,4)(range(16))
sage: f = a.minpoly(); f.factor()
x * (x^2 - 30*x - 80)
sage: f(a) == 0
True
sage: a = matrix(QQ, 4, [1..4^2])
sage: factor(a.minpoly())
x * (x^2 - 34*x - 80)
sage: factor(a.minpoly(’y’))
y * (y^2 - 34*y - 80)
345