Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 •num_pivots - The number of non-zero rows in the result, i.e. the rank. OUTPUT: A matrix in reduced row echelon form with num_pivots non-zero rows. If the base ring is ZZ or QQ then the entries are all integers. Note: It is easiest to use this function via a call to the random_matrix() function with the algorithm=’echelon_form’ keyword. We provide one example accessing this function directly, while the remainder will use this more general function. EXAMPLES: Matrices generated are in reduced row-echelon form with specified rank. If the base ring is QQ the result has only integer entries. sage: from sage.matrix.constructor import random_rref_matrix sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5, 6) sage: A=random_rref_matrix(matrix_space, num_pivots=4); A # random [ 1 0 0 -6 0 -3] [ 0 1 0 2 0 3] [ 0 0 1 -4 0 -2] [ 0 0 0 0 1 3] [ 0 0 0 0 0 0] sage: A.base_ring() Rational Field sage: (A.nrows(), A.ncols()) (5, 6) sage: A in sage.matrix.matrix_space.MatrixSpace(ZZ, 5, 6) True sage: A.rank() 4 sage: A==A.rref() True Matrices can be generated over other exact rings. sage: B=random_matrix(FiniteField(7), 4, 4, algorithm=’echelon_form’, num_pivots=3); B # random [1 0 0 0] [0 1 0 6] [0 0 1 4] [0 0 0 0] sage: B.rank() == 3 True sage: B.base_ring() Finite Field of size 7 sage: B==B.rref() True TESTS: Rank of a matrix must be an integer. sage: random_matrix(QQ, 120, 56, algorithm=’echelon_form’, num_pivots=61/2) Traceback (most recent call last): ... TypeError: the number of pivots must be an integer. Matrices must be generated over exact fields. 52 Chapter 2. Matrix Constructor
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
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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