Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
True
sage: matrix([[0,6],[0,0]]).is_symmetrizable(return_diag=True)
False
sage: matrix([2]).is_symmetrizable(positive=True)
True
sage: matrix([[1,2],[3,4]]).is_symmetrizable(return_diag=true)
[1, 2/3]
REFERENCES:
•[FZ2001] S. Fomin, A. Zelevinsky. Cluster Algebras 1: Foundations, arXiv:math/0104151 (2001).
iterates(v, n, rows=True)
Let A be this matrix and v be a free module element. If rows is True, return a matrix whose rows are the
entries of the following vectors:
v, vA, vA 2 , . . . , vA n−1 .
If rows is False, return a matrix whose columns are the entries of the following vectors:
INPUT:
•v - free module element
•n - nonnegative integer
EXAMPLES:
sage: A = matrix(ZZ,2, [1,1,3,5]); A
[1 1]
[3 5]
sage: v = vector([1,0])
sage: A.iterates(v,0)
[]
sage: A.iterates(v,5)
[ 1 0]
[ 1 1]
[ 4 6]
[ 22 34]
[124 192]
Another example:
sage: a = matrix(ZZ,3,range(9)); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: v = vector([1,0,0])
sage: a.iterates(v,4)
[ 1 0 0]
[ 0 1 2]
[ 15 18 21]
[180 234 288]
sage: a.iterates(v,4,rows=False)
[ 1 0 15 180]
[ 0 3 42 558]
[ 0 6 69 936]
v, Av, A 2 v, . . . , A n−1 v.
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