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 sage: random_matrix(QQ, 3, 4, algorithm=’echelonizable’, rank=-1) Traceback (most recent call last): ... ValueError: matrices must have rank zero or greater. sage: random_matrix(QQ, 3, 8, algorithm=’echelonizable’, rank=4) Traceback (most recent call last): ... ValueError: matrices cannot have rank greater than min(ncols,nrows). sage: random_matrix(QQ, 8, 3, algorithm=’echelonizable’, rank=4) Traceback (most recent call last): ... ValueError: matrices cannot have rank greater than min(ncols,nrows). The base ring must be exact. sage: random_matrix(RR, 3, 3, algorithm=’echelonizable’, rank=2) Traceback (most recent call last): ... TypeError: the base ring must be exact. Works for rank==1, too. sage: random_matrix( QQ, 3, 3, algorithm=’echelonizable’, rank=1).ncols() 3 AUTHOR: Billy Wonderly (2010-07) sage.matrix.constructor.random_matrix(ring, nrows, ncols=None, algorithm=’randomize’, *args, **kwds) This function is available as random_matrix(...) and matrix.random(...). Return a random matrix with entries in a specified ring, and possibly with additional properties. INPUT: •ring - base ring for entries of the matrix •nrows - Integer; number of rows •ncols - (default: None); number of columns; if None defaults to nrows •algorithm - (default: randomize); determines what properties the matrix will have. See examples below for possible additional arguments. –randomize - randomize the elements of the matrix, possibly controlling the density of non-zero entries. –echelon_form - creates a matrix in echelon form –echelonizable - creates a matrix that has a predictable echelon form –subspaces - creates a matrix whose four subspaces, when explored, have reasonably sized, integral valued, entries. –unimodular - creates a matrix of determinant 1. –diagonalizable - creates a diagonalizable matrix whose eigenvectors, if computed by hand, will have only integer entries. •*args, **kwds - arguments and keywords to describe additional properties. See more detailed documentation below. 46 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 Warning: An upper bound on the absolute value of the entries may be set when the algorithm is echelonizable or unimodular. In these cases it is possible for this constructor to fail with a ValueError. If you must have this routine return successfully, do not set upper_bound. This behavior can be partially controlled by a max_tries keyword. Note: When constructing matrices with random entries and no additional properties (i.e. when algorithm=’randomize’), most of the randomness is controlled by the random_element method for elements of the base ring of the matrix, so the documentation of that method may be relevant or useful. Also, the default is to not create zero entries, unless the density keyword is set to something strictly less than one. EXAMPLES: Random integer matrices. With no arguments, the majority of the entries are -1 and 1, never zero, and rarely “large.” sage: random_matrix(ZZ, 5, 5) [ -8 2 0 0 1] [ -1 2 1 -95 -1] [ -2 -12 0 0 1] [ -1 1 -1 -2 -1] [ 4 -4 -6 5 0] The distribution keyword set to uniform will limit values between -2 and 2, and never zero. sage: random_matrix(ZZ, 5, 5, distribution=’uniform’) [ 1 0 -2 1 1] [ 1 0 0 0 2] [-1 -2 0 2 -2] [-1 -1 1 1 2] [ 0 -2 -1 0 0] The x and y keywords can be used to distribute entries uniformly. When both are used x is the minimum and y is one greater than the the maximum. But still entries are never zero, even if the range contains zero. sage: random_matrix(ZZ, 4, 8, x=70, y=100) [81 82 70 81 78 71 79 94] [80 98 89 87 91 94 94 77] [86 89 85 92 95 94 72 89] [78 80 89 82 94 72 90 92] sage: random_matrix(ZZ, 3, 7, x=-5, y=5) [-3 3 1 -5 3 1 2] [ 3 3 0 3 -5 -2 1] [ 0 -2 -2 2 -3 -4 -2] If only x is given, then it is used as the upper bound of a range starting at 0. sage: random_matrix(ZZ, 5, 5, x=25) [20 16 8 3 8] [ 8 2 2 14 5] [18 18 10 20 11] [19 16 17 15 7] [ 0 24 3 17 24] To allow, and control, zero entries use the density keyword at a value strictly below the default of 1.0, even if distributing entries across an interval that does not contain zero already. Note that for a square matrix it is only necessary to set a single dimension. 47
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