random(Module,Module) -- get a random map of module

Synopsis

Function: random
Usage:

f = random(F, G)
Inputs:
- F, a module,
- G, a module,
Optional inputs:
- Density => a real number, default value 1, the proportion of entries to set
- MaximalRank => a Boolean value, default value false, whether to ensure that the resulting map has maximal rank: designed mostly for use with matrices of numbers: for polynomial rings, returns inhomogeneous results
- UpperTriangular => a Boolean value, default value false, whether to set just entries strictly above the diagonal
- Height => ..., default value 10, get a random homogeneous element from a graded ring
Outputs:
- f, a matrix, a random, graded, degree 0 map $f: G \to F$

Description

i1 : R = ZZ/101[x,y];

i2 : random(R^{1,2,3}, R^{1,2,3})

o2 = {-1} | 24             0       0   |
     {-2} | -29x+19y       -36     0   |
     {-3} | -29x2-8xy-22y2 19x-10y -30 |

             3      3
o2 : Matrix R  <-- R

i3 : random(ZZ^3, ZZ^10, Density => .3)

o3 = | 8 0 0 0 2 3 0 0 8 0 |
     | 0 5 7 5 0 6 0 3 6 0 |
     | 0 0 8 0 0 0 0 6 9 0 |

              3       10
o3 : Matrix ZZ  <-- ZZ

i4 : random(ZZ^3, ZZ^6, MaximalRank => true)

o4 = | 35047412   3247979   11405872  504891598   50976553853   5614985894  
     | 1157543751 107274048 376712434 16675528353 1683650456951 185451405627
     | 389888849  36132505  126885897 5616722964  567094366108  62464537627 
     ------------------------------------------------------------------------
     |
     |
     |

              3       6
o4 : Matrix ZZ  <-- ZZ

i5 : random(ZZ^6, ZZ^6, UpperTriangular => true)

o5 = | 0 2 5 3 7 5 |
     | 0 0 7 0 0 8 |
     | 0 0 0 1 2 9 |
     | 0 0 0 0 7 4 |
     | 0 0 0 0 0 3 |
     | 0 0 0 0 0 0 |

              6       6
o5 : Matrix ZZ  <-- ZZ

Caveat

Over a polynomial ring, specifying MaximalRank=>true will yield a non-homogeneous matrix.

Ways to use this method: