random(Module,Module) -- make a random module map

Synopsis

Usage:
f = random(F,G)
Function: random
Inputs:
- F, a module, a free module
- G, a module, a free module
Optional inputs:
- MaximalRank => 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
- Density => a real number, default value 1, the proportion of entries to set
- UpperTriangular => a Boolean value, default value false, whether to set just entries strictly above the diagonal
- Height => ..., -- random element of a type
Outputs:
- f, a matrix, a random, graded, degree 0 map, from G to F

Description

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

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

o2 = {-1} | -6       0        0   |
     {-2} | 25x-40y  33       0   |
     {-3} | 19xy+3y2 -50x+47y -11 |

             3       3
o2 : Matrix R  <--- R

i3 : random(ZZ^3,ZZ^6,MaximalRank=>true)

o3 = | 2111831491000  2191281489  12486458308 2192236654  24982626244 
     | 293695241100   304744459   1736508523  304877295   3474367377  
     | 16244362774094 16855497988 96046753438 16862845188 192168194047
     ------------------------------------------------------------------------
     262608779283  |
     36521355551   |
     2020005997881 |

              3        6
o3 : Matrix ZZ  <--- ZZ

i4 : random(ZZ^3,ZZ^10,Density=>.3)

o4 = | 0 8 6 0 1 0 0 0 0 5 |
     | 1 0 0 6 0 5 0 0 0 0 |
     | 0 0 0 0 0 4 0 8 6 0 |

              3        10
o4 : Matrix ZZ  <--- ZZ

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

o5 = | 0 4 4 5 1 1 |
     | 0 0 6 5 2 4 |
     | 0 0 0 6 5 6 |
     | 0 0 0 0 6 7 |
     | 0 0 0 0 0 4 |
     | 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.

random(Module,Module) -- make a random module map

Synopsis

Description

Caveat

See also