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
Outputs:
- f, a matrix, a random, graded, degree 0 map, from G to F
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

Description

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

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

o2 = {-1} | 13              0     0 |
     {-2} | -14x+37y        30    0 |
     {-3} | -35x2-19xy-33y2 x+30y 2 |

             3       3
o2 : Matrix R  <--- R

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

o3 = | 8877984074  9450313677  6820199200  490543164404  150139209 
     | 61657744706 65632583209 47366395078 3406830307932 1042719278
     | 9002128113  9582460817  6915568495  497402606129  152238663 
     ------------------------------------------------------------------------
     61133143819  |
     424570684685 |
     61987990583  |

              3        6
o3 : Matrix ZZ  <--- ZZ

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

o4 = | 3 3 0 0 0 9 8 0 4 0 |
     | 0 8 6 0 0 8 3 6 0 0 |
     | 8 0 9 9 0 0 6 0 0 1 |

              3        10
o4 : Matrix ZZ  <--- ZZ

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

o5 = | 0 9 2 4 1 9 |
     | 0 0 9 6 9 9 |
     | 0 0 0 0 8 9 |
     | 0 0 0 0 9 5 |
     | 0 0 0 0 0 6 |
     | 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