symbolicSlackMatrix -- computes the symbolic slack matrix

Usage:

S = symbolicSlackMatrix V

S = symbolicSlackMatrix X

S = symbolicSlackMatrix M

S = symbolicSlackMatrix P

S = symbolicSlackMatrix C
Inputs:
- V, a list, a list of points treated as the vertices of polytope; or vectors of a linear matroid if matroid is given as Object; or a list of indices comprising the facets of an abstract polytope if abstractPolytope is given as Object.
- X, a matrix, a slack matrix
- M, a matroid, a matroid
- P, a convex polyhedron, a polytope
- C, a convex rational cone, a cone
Optional inputs:
- CoefficientRing => ..., default value QQ, specifies the coefficient ring of the underlying ring of the matrix
- Object => ..., default value "polytope", specify combinatorial object
- Vars => ..., default value {}, specifies the variables to use to create the underlying ring of the matrix
Outputs:
- S, a matrix, the symbolic slack matrix with indexed variables in symbol x

Description

The symbolic slack matrix records the combinatorial structure of the given object. Its (i, j)-entry is 0 if element i is in hyperplane j and it is a variable otherwise. Variables are indexed left to right by rows.

i1 : V = {{0, 0}, {0, 1}, {1, 1}, {1, 0}};

i2 : S = symbolicSlackMatrix V

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}

o2 = | 0   x_0 0   x_1 |
     | x_2 0   0   x_3 |
     | 0   x_4 x_5 0   |
     | x_6 0   x_7 0   |

                        4                 4
o2 : Matrix (QQ[x ..x ])  <-- (QQ[x ..x ])
                 0   7             0   7

i3 : M = matroid({0, 1, 2, 3, 4, 5}, {{1, 2, 3}, {0, 2, 4}, {0, 3, 5}, {1, 4, 5}}, EntryMode => "nonbases");

i4 : S = symbolicSlackMatrix M

o4 = | x_0  0    x_1  x_2  0    x_3  0    |
     | 0    x_4  x_5  x_6  x_7  0    0    |
     | x_8  x_9  0    x_10 0    0    x_11 |
     | x_12 0    x_13 0    x_14 0    x_15 |
     | 0    x_16 x_17 0    0    x_18 x_19 |
     | 0    0    0    x_20 x_21 x_22 x_23 |

                         6                  7
o4 : Matrix (QQ[x ..x  ])  <-- (QQ[x ..x  ])
                 0   23             0   23

i5 : V = {{1, 2, 3}, {4, 5, 6}, {1, 2, 4, 5}, {1, 3, 4, 6}, {2, 3, 5, 6}};

i6 : S = symbolicSlackMatrix(V, Object => "abstractPolytope")

o6 = | 0    x_0 0    0   x_1 |
     | 0    x_2 0    x_3 0   |
     | 0    x_4 x_5  0   0   |
     | x_6  0   0    0   x_7 |
     | x_8  0   0    x_9 0   |
     | x_10 0   x_11 0   0   |

                         6                  5
o6 : Matrix (QQ[x ..x  ])  <-- (QQ[x ..x  ])
                 0   11             0   11

Ways to use symbolicSlackMatrix:

symbolicSlackMatrix(Cone)
symbolicSlackMatrix(List)
symbolicSlackMatrix(Matrix)
symbolicSlackMatrix(Matroid)
symbolicSlackMatrix(Polyhedron)

For the programmer

The object symbolicSlackMatrix is a method function with options.

The source of this document is in SlackIdeals.m2:1803:0.

symbolicSlackMatrix -- computes the symbolic slack matrix

Description

See also

Ways to use symbolicSlackMatrix:

For the programmer