Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
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sage: P.is_sparse()
False
symplectic_form()
Find a symplectic basis for self if self is an anti-symmetric, alternating matrix.
Returns a pair (F, C) such that the rows of C form a symplectic basis for self and F = C * self *
C.transpose().
Raises a ValueError if self is not anti-symmetric, or self is not alternating.
Anti-symmetric means that M = −M t . Alternating means that the diagonal of M is identically zero.
A symplectic basis is a basis of the form e 1 , . . . , e j , f 1 , . . . f j , z 1 , . . . , z k such that
•z i Mv t = 0 for all vectors v
•e i Me j t = 0 for all i, j
•f i Mf j t = 0 for all i, j
•e i Mf i t = 1 for all i
•e i Mf j t = 0 for all i not equal j.
The ordering for the factors d i |d i+1 and for the placement of zeroes was chosen to agree with the output
of smith_form.
See the example for a pictorial description of such a basis.
EXAMPLES:
sage: E = matrix(ZZ, 5, 5, [0, 14, 0, -8, -2, -14, 0, -3, -11, 4, 0, 3, 0, 0, 0, 8, 11, 0, 0
[ 0 14 0 -8 -2]
[-14 0 -3 -11 4]
[ 0 3 0 0 0]
[ 8 11 0 0 8]
[ 2 -4 0 -8 0]
sage: F, C = E.symplectic_form()
sage: F
[ 0 0 1 0 0]
[ 0 0 0 2 0]
[-1 0 0 0 0]
[ 0 -2 0 0 0]
[ 0 0 0 0 0]
sage: F == C * E * C.transpose()
True
sage: E.smith_form()[0]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 0]
transpose()
Returns the transpose of self, without changing self.
EXAMPLES:
We create a matrix, compute its transpose, and note that the original matrix is not changed.
336 Chapter 17. Dense matrices over the integer ring