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} | 11x-31y        42      0  |
     {-3} | -32x2-5xy+41y2 31x+14y -1 |

             3       3
o2 : Matrix R  <--- R

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

o3 = | 3455790627 3560647429 648127310 22209071949 17908941594 241347 |
     | 949988836  978813582  178168705 6105221548  4923125819  66347  |
     | 418983217  431696167  78579547  2692648292  2171296539  29261  |

              3        6
o3 : Matrix ZZ  <--- ZZ

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

o4 = | 0 0 0 4 0 8 0 0 5 0 |
     | 0 0 0 0 0 0 2 1 0 2 |
     | 0 8 0 0 0 0 0 0 0 8 |

              3        10
o4 : Matrix ZZ  <--- ZZ

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

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