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ExtraZeroRows -- enlarging a 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 object typically live in the space $\RR^{2 \times 4}$. However, by adding an additional row, the objects live in $\RR^{3 \times 4} \cong \RR^{12}$. Reading the down the entries of columns corresponds to reading row-by-row the entries of the corresponding matrix.

See also

Functions with optional argument named ExtraZeroRows :

For the programmer

The object ExtraZeroRows is a symbol.

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