Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•Rob Beezer (2011-05-20)
decomposition(algorithm=’spin’, is_diagonalizable=False, dual=False)
Returns the decomposition of the free module on which this matrix A acts from the right (i.e., the action is
x goes to x A), along with whether this matrix acts irreducibly on each factor. The factors are guaranteed
to be sorted in the same way as the corresponding factors of the characteristic polynomial.
Let A be the matrix acting from the on the vector space V of column vectors. Assume that A is square.
This function computes maximal subspaces W_1, ..., W_n corresponding to Galois conjugacy classes of
eigenvalues of A. More precisely, let f(X) be the characteristic polynomial of A. This function computes
the subspace W i = ker(g ( A) n ), where g i (X) is an irreducible factor of f(X) and g i (X) exactly divides
f(X). If the optional parameter is_diagonalizable is True, then we let W i = ker(g(A)), since then we
know that ker(g(A)) = ker(g(A) n ).
INPUT:
•self - a matrix
•algorithm - ‘spin’ (default): algorithm involves iterating the action of self on a vector. ‘kernel’:
naively just compute ker(f i (A)) for each factor f i .
•dual - bool (default: False): If True, also returns the corresponding decomposition of V under the
action of the transpose of A. The factors are guaranteed to correspond.
•is_diagonalizable - if the matrix is known to be diagonalizable, set this to True, which might
speed up the algorithm in some cases.
Note: If the base ring is not a field, the kernel algorithm is used.
OUTPUT:
•Sequence - list of pairs (V,t), where V is a vector spaces and t is a bool, and t is True exactly when
the charpoly of self on V is irreducible.
•(optional) list - list of pairs (W,t), where W is a vector space and t is a bool, and t is True exactly when
the charpoly of the transpose of self on W is irreducible.
EXAMPLES:
sage: A = matrix(ZZ, 4, [3,4,5,6,7,3,8,10,14,5,6,7,2,2,10,9])
sage: B = matrix(QQ, 6, range(36))
sage: B*11
[ 0 11 22 33 44 55]
[ 66 77 88 99 110 121]
[132 143 154 165 176 187]
[198 209 220 231 242 253]
[264 275 286 297 308 319]
[330 341 352 363 374 385]
sage: A.decomposition()
[
(Ambient free module of rank 4 over the principal ideal domain Integer Ring, True)
]
sage: B.decomposition()
[
(Vector space of degree 6 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1 -2 -3 -4]
[ 0 1 2 3 4 5], True),
(Vector space of degree 6 and dimension 4 over Rational Field
Basis matrix:
147