Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
OUTPUT:
•U - unimodular transformation matrix such that U.transpose() * M * U is LLL-reduced.
ALGORITHM: Use PARI
EXAMPLES:
sage: M = Matrix(ZZ, 2, 2, [5,3,3,2]) ; M
[5 3]
[3 2]
sage: U = M.LLL_gram(); U
[-1 1]
[ 1 -2]
sage: U.transpose() * M * U
[1 0]
[0 1]
Semidefinite and indefinite forms no longer raise a ValueError:
sage: Matrix(ZZ,2,2,[2,6,6,3]).LLL_gram()
[-3 -1]
[ 1 0]
sage: Matrix(ZZ,2,2,[1,0,0,-1]).LLL_gram()
[ 0 -1]
[ 1 0]
antitranspose()
Returns the antitranspose of self, without changing self.
EXAMPLES:
sage: A = matrix(2,3,range(6))
sage: type(A)
sage: A.antitranspose()
[5 2]
[4 1]
[3 0]
sage: A
[0 1 2]
[3 4 5]
sage: A.subdivide(1,2); A
[0 1|2]
[---+-]
[3 4|5]
sage: A.antitranspose()
[5|2]
[-+-]
[4|1]
[3|0]
augment(right, subdivide=False)
Returns a new matrix formed by appending the matrix (or vector) right on the right side of self.
INPUT:
•right - a matrix, vector or free module element, whose dimensions are compatible with self.
•subdivide - default: False - request the resulting matrix to have a new subdivision, separating
self from right.
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