Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: t.maxspin(v)
[(1, 0, 0), (0, 1, 2), (15, 18, 21)]
sage: k = t.kernel(); k
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2 1]
sage: t.maxspin(k.0)
[(1, -2, 1)]
minimal_polynomial(var=’x’, **kwds)
This is a synonym for self.minpoly
EXAMPLES:
sage: a = matrix(QQ, 4, range(16))
sage: a.minimal_polynomial(’z’)
z^3 - 30*z^2 - 80*z
sage: a.minpoly()
x^3 - 30*x^2 - 80*x
minors(k)
Return the list of all k × k minors of self.
Let A be an m × n matrix and k an integer with 0 ≤ k, k ≤ m and k ≤ n. A k × k minor of A is the
determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. There are no
k × k minors of A if k is larger than either m or n.
The returned list is sorted in lexicographical row major ordering, e.g., if A is a 3×3 matrix then the minors
returned are with these rows/columns: [ [0, 1]x[0, 1], [0, 1]x[0, 2], [0, 1]x[1, 2], [0, 2]x[0, 1], [0, 2]x[0, 2],
[0, 2]x[1, 2], [1, 2]x[0, 1], [1, 2]x[0, 2], [1, 2]x[1, 2] ].
INPUT:
•k – integer
EXAMPLES:
sage: A = Matrix(ZZ,2,3,[1,2,3,4,5,6]); A
[1 2 3]
[4 5 6]
sage: A.minors(2)
[-3, -6, -3]
sage: A.minors(1)
[1, 2, 3, 4, 5, 6]
sage: A.minors(0)
[1]
sage: A.minors(5)
[]
sage: k = GF(37)
sage: P. = PolynomialRing(k)
sage: A = Matrix(P,2,3,[x0*x1, x0, x1, x2, x2 + 16, x2 + 5*x1 ])
sage: A.minors(2)
[x0*x1*x2 + 16*x0*x1 - x0*x2, 5*x0*x1^2 + x0*x1*x2 - x1*x2, 5*x0*x1 + x0*x2 - x1*x2 - 16*x1]
minpoly(var=’x’, **kwds)
Return the minimal polynomial of self.
This uses a simplistic - and potentially very very slow - algorithm that involves computing kernels to
determine the powers of the factors of the charpoly that divide the minpoly.
214 Chapter 7. Base class for matrices, part 2