Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
but yet the diagonal cannot be completely arbitrary in this case:
sage: n = matrix(Zmod(4), [[0, 1], [-1, 3]])
sage: n.is_skew_symmetric()
False
is_skew_symmetrizable(return_diag=False, positive=True)
This function takes a square matrix over an ordered integral domain and checks if it is skew-symmetrizable.
A matrix B is skew-symmetrizable iff there exists an invertible diagonal matrix D such that DB is skewsymmetric.
Warning: Expects self to be a matrix over an ordered integral domain.
INPUT:
•return_diag – bool(default:False) if True and self is skew-symmetrizable the diagonal entries
of the matrix D are returned.
•positive – bool(default:True) if True, the condition that D has positive entries is added.
OUTPUT:
•True – if self is skew-symmetrizable and return_diag is False
•the diagonal entries of a matrix D such that DB is skew-symmetric – iff self is skew-symmetrizable
and return_diag is True
•False – iff self is not skew-symmetrizable
EXAMPLES:
sage: matrix([[0,6],[3,0]]).is_skew_symmetrizable(positive=False)
True
sage: matrix([[0,6],[3,0]]).is_skew_symmetrizable(positive=True)
False
sage: M = matrix(4,[0,1,0,0,-1,0,-1,0,0,2,0,1,0,0,-1,0]); M
[ 0 1 0 0]
[-1 0 -1 0]
[ 0 2 0 1]
[ 0 0 -1 0]
sage: M.is_skew_symmetrizable(return_diag=True)
[1, 1, 1/2, 1/2]
sage: M2 = diagonal_matrix([1,1,1/2,1/2])*M; M2
[ 0 1 0 0]
[ -1 0 -1 0]
[ 0 1 0 1/2]
[ 0 0 -1/2 0]
sage: M2.is_skew_symmetric()
True
REFERENCES:
•[FZ2001] S. Fomin, A. Zelevinsky. Cluster Algebras 1: Foundations, arXiv:math/0104151 (2001).
is_sparse()
Return True if this is a sparse matrix.
In Sage, being sparse is a property of the underlying representation, not the number of nonzero entries.
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