ExtraZeroRows -- Enlarge a (weight) matrix with zero rows

Description

The option ExtraZeroRows is used by the functions matchingFieldPolytope and weightMatrixCone. In each case, the option controls the ambient space of the polyhedron. By default the value is zero. It is typically used internally for computing Minkowski sums of polyhedra that would ordinarily belong to different ambient spaces.

i1 : L = diagonalMatchingField(2, 4)

o1 = Grassmannian Matching Field for Gr(2, 4)

o1 : GrMatchingField

i2 : P = matchingFieldPolytope(L, ExtraZeroRows => 1)

o2 = P

o2 : Polyhedron

i3 : vertices P

o3 = | 1 1 0 1 0 0 |
     | 0 0 1 0 1 0 |
     | 0 0 0 0 0 1 |
     | 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 |
     | 1 0 0 0 0 0 |
     | 0 1 1 0 0 0 |
     | 0 0 0 1 1 1 |
     | 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 |

              12       6
o3 : Matrix QQ   <-- QQ

i4 : C = weightMatrixCone(L, ExtraZeroRows => 1)

o4 = C

o4 : Cone

i5 : rays C

o5 = | 0  0  0  |
     | 0  0  0  |
     | 0  0  0  |
     | 0  0  0  |
     | 0  0  0  |
     | 0  0  -1 |
     | 0  -1 -1 |
     | -1 -1 -1 |
     | 0  0  0  |
     | 0  0  0  |
     | 0  0  0  |
     | 0  0  0  |

              12       3
o5 : Matrix ZZ   <-- ZZ

In the above examples, the polyhedral objects typically live in the space $\RR^{2 \times 4}$. However, by adding an extra row, the objects live in $\RR^{3 \times 4} \cong \RR^{12}$. Reading the entries of columns from top-to-bottom corresponds to reading the entries of the corresponding matrix row-by-row.

Functions with optional argument named ExtraZeroRows:

matchingFieldPolytope(...,ExtraZeroRows=>...) -- see matchingFieldPolytope -- The polytope of a matching field
weightMatrixCone(...,ExtraZeroRows=>...) -- see weightMatrixCone -- The cone of weight matrices that induce the matching field

For the programmer

The object ExtraZeroRows is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MatchingFields.m2:2096:0.

ExtraZeroRows -- Enlarge a (weight) matrix with zero rows

Description

See also

Menu

Functions with optional argument named ExtraZeroRows:

For the programmer