Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[ 1 0 0 0 -5 4]
[ 0 1 0 0 -4 3]
[ 0 0 1 0 -3 2]
[ 0 0 0 1 -2 1], False)
]
decomposition_of_subspace(M, check_restrict=True, **kwds)
Suppose the right action of self on M leaves M invariant. Return the decomposition of M as a list of pairs
(W, is_irred) where is_irred is True if the charpoly of self acting on the factor W is irreducible.
Additional inputs besides M are passed onto the decomposition command.
INPUT:
•M – A subspace of the free module self acts on.
•check_restrict – A boolean (default: True); Call restrict with or without check.
•kwds – Keywords that will be forwarded to decomposition().
EXAMPLES:
sage: t = matrix(QQ, 3, [3, 0, -2, 0, -2, 0, 0, 0, 0]); t
[ 3 0 -2]
[ 0 -2 0]
[ 0 0 0]
sage: t.fcp(’X’) # factored charpoly
(X - 3) * X * (X + 2)
sage: v = kernel(t*(t+2)); v # an invariant subspace
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[0 1 0]
[0 0 1]
sage: D = t.decomposition_of_subspace(v); D
[
(Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[0 0 1], True),
(Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[0 1 0], True)
]
sage: t.restrict(D[0][0])
[0]
sage: t.restrict(D[1][0])
[-2]
We do a decomposition over ZZ:
sage: a = matrix(ZZ,6,[0, 0, -2, 0, 2, 0, 2, -4, -2, 0, 2, 0, 0, 0, -2, -2, 0, 0, 2, 0, -2,
sage: a.decomposition_of_subspace(ZZ^6)
[
(Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 0 1 -1 1 -1]
[ 0 1 0 -1 2 -1], False),
(Free module of degree 6 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1 0 -1 0 1 0]
[ 0 1 0 0 0 0]
[ 0 0 0 1 0 0]
148 Chapter 7. Base class for matrices, part 2