Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: random_matrix(RR, 40, 88, algorithm=’echelon_form’, num_pivots=39)
Traceback (most recent call last):
...
TypeError: the base ring must be exact.
Matrices must have the number of pivot columns be less than or equal to the number of rows.
sage: C=random_matrix(ZZ, 6,4, algorithm=’echelon_form’, num_pivots=7); C
Traceback (most recent call last):
...
ValueError: number of pivots cannot exceed the number of rows or columns.
Matrices must have the number of pivot columns be less than or equal to the number of columns.
sage: D=random_matrix(QQ, 1,3, algorithm=’echelon_form’, num_pivots=5); D
Traceback (most recent call last):
...
ValueError: number of pivots cannot exceed the number of rows or columns.
Matrices must have the number of pivot columns be greater than zero.
sage: random_matrix(QQ, 5, 4, algorithm=’echelon_form’, num_pivots=-1)
Traceback (most recent call last):
...
ValueError: the number of pivots must be zero or greater.
AUTHOR:
Billy Wonderly (2010-07)
sage.matrix.constructor.random_subspaces_matrix(parent, rank=None)
This function is available as random_subspaces_matrix(...) and matrix.random_subspaces(...).
Create a matrix of the designated size and rank whose right and left null spaces, column space, and row space
have desirable properties that simplify the subspaces.
INPUT:
•parent - A matrix space specifying the base ring, dimensions, and representation (dense/sparse) for the
result. The base ring must be exact.
•rank - The desired rank of the return matrix (default: None).
OUTPUT:
A matrix whose natrual basis vectors for its four subspaces, when computed, have reasonably sized, integral
valued, entries.
Note: It is easiest to use this function via a call to the random_matrix() function with the
algorithm=’subspaces’ keyword. We provide one example accessing this function directly, while the
remainder will use this more general function.
EXAMPLES:
A 6x8 matrix with designated rank of 3. The four subspaces are determined using one simple routine in which
we augment the original matrix with the equal row dimension identity matrix. The resulting matrix is then put in
reduced row-echelon form and the subspaces can then be determined by analyzing subdivisions of this matrix.
See the four subspaces routine in [BEEZER] for more.
53