Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: random_matrix(QQ,12,algorithm=’diagonalizable’,eigenvalues=[4,2,6],dimensions=[2,3,5,2])
Traceback (most recent call last):
...
ValueError: each eigenvalue must have a corresponding dimension and each dimension a correspondi
TODO:
Modify the routine to allow for complex eigenvalues.
AUTHOR:
Billy Wonderly (2010-07)
sage.matrix.constructor.random_echelonizable_matrix(parent, rank, upper_bound=None,
max_tries=100)
This function is available as random_echelonizable_matrix(...) and matrix.random_echelonizable(...).
Generate a matrix of a desired size and rank, over a desired ring, whose reduced row-echelon form has only
integral values.
INPUT:
•parent - A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the
result. The base ring must be exact.
•rank - Rank of result, i.e the number of non-zero rows in the reduced row echelon form.
•upper_bound - If designated, size control of the matrix entries is desired. Set upper_bound to 1
more than the maximum value entries can achieve. If None, no size control occurs. But see the warning
below. (default: None)
•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)
OUTPUT:
A matrix not in reduced row-echelon form with the desired dimensions and properties.
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=’echelonizable’ keyword. We provide one example accessing this function directly, while
the remainder will use this more general function.
EXAMPLES:
Generated matrices have the desired dimensions, rank and entry size. The matrix in reduced row-echelon form
has only integer entries.
sage: from sage.matrix.constructor import random_echelonizable_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5, 6)
sage: A=random_echelonizable_matrix(matrix_space, rank=4, upper_bound=40); A # random
[ 1 -1 1 -3 -4 6]
[ 5 -4 0 8 4 19]
[ -3 3 -2 4 7 -16]
[ -4 5 -7 26 31 -31]
44 Chapter 2. Matrix Constructor