Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage.matrix.constructor.random_unimodular_matrix(parent, upper_bound=None,
max_tries=100)
This function is available as random_unimodular_matrix(...) and matrix.random_unimodular(...).
Generate a random unimodular (determinant 1) matrix of a desired size over a desired ring.
INPUT:
•parent - A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the
result. The base ring must be exact.
•upper_bound - For large matrices over QQ or ZZ, upper_bound is the largest value matrix entries
can achieve. But see the warning below.
•max_tries - If designated, number of tries used to generate each new random row; only matters when
upper_bound!=None. Used to prevent endless looping. (default: 100)
A matrix not in reduced row-echelon form with the desired dimensions and properties.
OUTPUT:
An invertible matrix with the desired properties and determinant 1.
Warning: When upper_bound is set, it is possible for this constructor to fail with a ValueError. This
may happen when the upper_bound, rank and/or matrix dimensions are all so small that it becomes
infeasible or unlikely to create the requested matrix. If you must have this routine return successfully, do not
set upper_bound.
Note: It is easiest to use this function via a call to the random_matrix() function with the
algorithm=’unimodular’ keyword. We provide one example accessing this function directly, while
the remainder will use this more general function.
EXAMPLES:
A matrix size 5 over QQ.
sage: from sage.matrix.constructor import random_unimodular_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5)
sage: A=random_unimodular_matrix(matrix_space); A # random
[ -8 31 85 148 -419]
[ 2 -9 -25 -45 127]
[ -1 10 30 65 -176]
[ -3 12 33 58 -164]
[ 5 -21 -59 -109 304]
sage: det(A)
1
A matrix size 6 with entries no larger than 50.
sage: B=random_matrix(ZZ, 7, algorithm=’unimodular’, upper_bound=50);B # random
[ -1 0 3 1 -2 2 9]
[ 1 2 -5 0 14 19 -49]
[ -3 -2 12 5 -6 -4 24]
[ 1 2 -9 -3 3 4 -7]
[ -2 -1 7 2 -8 -5 31]
[ 2 2 -6 -3 8 16 -32]
[ 1 2 -9 -2 5 6 -12]
sage: det(B)
1
56 Chapter 2. Matrix Constructor