Sage Reference Manual: Matrices and Spaces of Matrices

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2Warning: An upper bound on the absolute value of the entries may be set when the algorithmis echelonizable or unimodular. In these cases it is possible for this constructor to fail with aValueError. If you must have this routine return successfully, do not set upper_bound. This behaviorcan be partially controlled by a max_tries keyword.Note: When constructing matrices with random entries and no additional properties (i.e. whenalgorithm=’randomize’), most of the randomness is controlled by the random_element method forelements 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 yis 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 ifdistributing entries across an interval that does not contain zero already. Note that for a square matrix it is onlynecessary to set a single dimension.47