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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An example over Q:
sage: A = MatrixSpace(QQ,3)(range(9))
sage: A.charpoly(’x’)
x^3 - 12*x^2 - 18*x
sage: A.trace()
12
sage: A.determinant()
0
We compute the characteristic polynomial of a matrix over the polynomial ring Z[a]:
sage: R. = PolynomialRing(ZZ)
sage: M = MatrixSpace(R,2)([a,1, a,a+1]); M
[ a 1]
[ a a + 1]
sage: f = M.charpoly(’x’); f
x^2 + (-2*a - 1)*x + a^2
sage: f.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in a over Integer Ring
sage: M.trace()
2*a + 1
sage: M.determinant()
a^2
We compute the characteristic polynomial of a matrix over the multi-variate polynomial ring Z[x, y]:
sage: R. = PolynomialRing(ZZ,2)
sage: A = MatrixSpace(R,2)([x, y, x^2, y^2])
sage: f = A.charpoly(’x’); f
x^2 + (-y^2 - x)*x - x^2*y + x*y^2
It’s a little difficult to distinguish the variables. To fix this, we temporarily view the indeterminate as Z:
sage: with localvars(f.parent(), ’Z’): print f
Z^2 + (-y^2 - x)*Z - x^2*y + x*y^2
We could also compute f in terms of Z from the start:
sage: A.charpoly(’Z’)
Z^2 + (-y^2 - x)*Z - x^2*y + x*y^2
Here is an example over a number field:
sage: x = QQ[’x’].gen()
sage: K. = NumberField(x^2 - 2)
sage: m = matrix(K, [[a-1, 2], [a, a+1]])
sage: m.charpoly(’Z’)
Z^2 - 2*a*Z - 2*a + 1
sage: m.charpoly(’a’)(m) == 0
True
Here is an example over a general commutative ring, that is to say, as of version 4.0.2, SAGE does not
even positively determine that S in the following example is an integral domain. But the computation of
the characteristic polynomial succeeds as follows:
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