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} | -21            0        0  |
     {-2} | 38x+9y         42       0  |
     {-3} | -4x2+20xy-24y2 -44x+13y 24 |

             3       3
o2 : Matrix R  <--- R

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

o3 = | 40928540 907344766 499573195940 13292283796 3655966071 95443725397 |
     | 2542169  56357382  31029700981  825615936   227080910  5928240909  |
     | 5759554  127683546 70301036120  1870519337  514475566  13431050427 |

              3        6
o3 : Matrix ZZ  <--- ZZ

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

o4 = | 0 0 0 0 0 0 0 0 0 0 |
     | 3 3 0 0 8 0 0 0 0 0 |
     | 0 0 0 0 4 0 0 0 0 4 |

              3        10
o4 : Matrix ZZ  <--- ZZ

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

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