Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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charpoly(var=’x’, algorithm=’linbox’)
Return the characteristic 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.charpoly(); f
x^3 - 7/12*x^2 - 149/40*x + 97/30
sage: f(a)
[0 0 0]
[0 0 0]
[0 0 0]
TESTS:
The cached polynomial should be independent of the var argument (trac ticket #12292). We check (indirectly)
that the second call uses the cached value by noting that its result is not cached:
sage: M = MatrixSpace(QQ, 2)
sage: A = M(range(0, 2^2))
sage: type(A)
sage: A.charpoly(’x’)
x^2 - 3*x - 2
sage: A.charpoly(’y’)
y^2 - 3*y - 2
sage: A._cache[’charpoly_linbox’]
x^2 - 3*x - 2
column(i, from_list=False)
Return the i-th column of this matrix as a dense vector.
INPUT:
• i - integer
• from_list - ignored
EXAMPLES:
sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(1)
(-2/3, 4/9)
sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(1,from_list=True)
(-2/3, 4/9)
sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).column(-1)
(-2/3, 4/9)
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