maxMinor -- Returns a maximal minor of the matrix of full rank.

Synopsis

• Usage:

  MM = maxMinor(Mat)

• Inputs:
  • m, a matrix
• Optional inputs:
  • Strategy => ..., default value null, choose between Exact and Numeric algorithms
• Outputs:
  • a matrix

Description

From a given m x n - Matrix of rank r, maxMinor returns an r x r full rank Matrix. This method uses twice the method maxCol by transposing twice.

i1 : M=matrix {{1,2,3},{1,2,3},{4,5,6},{4,5,6}}

o1 = | 1 2 3 |
     | 1 2 3 |
     | 4 5 6 |
     | 4 5 6 |

              4        3
o1 : Matrix ZZ  <--- ZZ

i2 : maxMinor M

o2 = | 1 2 |
     | 4 5 |

              2        2
o2 : Matrix ZZ  <--- ZZ

NOTE: because of the necessity of rank the base field need to be QQ for doing generic evaluation. If not, one gets the message: expected an affine ring (consider Generic=>true to work over QQ).

i3 : R=QQ[a..g]

o3 = R

o3 : PolynomialRing

i4 : M=matrix {{a,a,b},{c,c,d},{e,e,f},{g,g,g}}

o4 = | a a b |
     | c c d |
     | e e f |
     | g g g |

             4       3
o4 : Matrix R  <--- R

i5 : maxMinor M

o5 = | a b |
     | c d |

             2       2
o5 : Matrix R  <--- R

See also

• maxCol -- Returns a submatrix form by a maximal set of linear independent columns.
• rank -- compute the rank