getSubmatrixOfRank -- tries to find a submatrix of the given rank

Synopsis

• Usage:

  getSubmatrixOfRank(n1, M1)

• Inputs:
  • n1, an integer,
  • M1, a matrix,
• Optional inputs:
  • DetStrategy => ..., default value Rank, DetStrategy is a strategy for allowing the user to choose how determinants (or rank), is computed
  • MaxMinors => ..., default value null, an option to control depth of search
  • PointOptions => ..., default value {Strategy => Default, Homogeneous => false, Replacement => Binomial, ExtendField => true, PointCheckAttempts => 0, DecompositionStrategy => Decompose, NumThreadsToUse => 1, DimensionFunction => dim, Verbose => false}, options to pass to functions in the package RandomPoints
  • Strategy => ..., default value new OptionTable from {Points => 0, Random => 0, GRevLexLargest => 0, LexSmallestTerm => 25, LexLargest => 0, LexSmallest => 25, GRevLexSmallestTerm => 25, RandomNonzero => 0, GRevLexSmallest => 25}, strategies for choosing submatrices
  • Threads => ..., default value 1
  • Verbose => ..., default value false
• Outputs:
  • a list, the first entry is a list of row indices, the second is a list of column indices

Description

i1 : R = QQ[x,y];

i2 : M = matrix{{x,y,2,0,2*x+y}, {0,0,1,0,x}, {x,y,0,0,y}};

             3      5
o2 : Matrix R  <-- R

i3 : l = getSubmatrixOfRank(2, M)

o3 = {{1, 2}, {2, 1}}

o3 : List

i4 : (M^(l#0))_(l#1)

o4 = | 1 0 |
     | 0 y |

             2      2
o4 : Matrix R  <-- R

i5 : l = getSubmatrixOfRank(2, M)

o5 = {{1, 0}, {2, 1}}

o5 : List

i6 : (M^(l#0))_(l#1)

o6 = | 1 0 |
     | 2 y |

             2      2
o6 : Matrix R  <-- R

i7 : getSubmatrixOfRank(3, M)

See also

• isRankAtLeast -- determines if the matrix has rank at least a number
• getSubmatrixOfRank(...,Strategy=>...) -- strategies for choosing submatrices