markovMatrices -- The matrices whose minors form the ideal associated to the list of independence statements of the graph.

Synopsis

Usage:
markovMatrices(R,G,S)
Inputs:
- R, a ring, R must be a markovRing
- G, an object of class Digraph, directed acyclic graph
- S, a list, a list of independence statements that are true for the DAG G
Outputs:
- a list, whose elements are instances of Matrix. Minors of these matrices form the independence ideal for the independent statements of the Digraph.

Description

List of matrices encoding the independent statements of the Digraph G. The 2x2 minors of each matrix generate the ideal of independence constraints of the Digraph G. the This method is used in markovIdeals. But it is exported to be able to see constraints not as polynomials but as minors of matrices in this list.

i1 : G = digraph { {1, {2,3}}, {2, {4}}, {3, {4}} }

o1 = Digraph{1 => set {2, 3}}
             2 => set {4}
             3 => set {4}
             4 => set {}

o1 : Digraph

i2 : S = localMarkov G

o2 = {{{2}, {3}, {1}}, {{1}, {4}, {2, 3}}}

o2 : List

i3 : R = markovRing (4:2)

o3 = R

o3 : PolynomialRing

i4 : L = markovMatrices (R,G,S)

o4 = {| p_(1,1,1,1)+p_(1,1,1,2) p_(1,1,2,1)+p_(1,1,2,2) |, |
      | p_(1,2,1,1)+p_(1,2,1,2) p_(1,2,2,1)+p_(1,2,2,2) |  |
     ------------------------------------------------------------------------
     p_(2,1,1,1)+p_(2,1,1,2) p_(2,1,2,1)+p_(2,1,2,2) |, | p_(1,1,1,1)
     p_(2,2,1,1)+p_(2,2,1,2) p_(2,2,2,1)+p_(2,2,2,2) |  | p_(2,1,1,1)
     ------------------------------------------------------------------------
     p_(1,1,1,2) |, | p_(1,1,2,1) p_(1,1,2,2) |, | p_(1,2,1,1) p_(1,2,1,2) |,
     p_(2,1,1,2) |  | p_(2,1,2,1) p_(2,1,2,2) |  | p_(2,2,1,1) p_(2,2,1,2) | 
     ------------------------------------------------------------------------
     | p_(1,2,2,1) p_(1,2,2,2) |}
     | p_(2,2,2,1) p_(2,2,2,2) |

o4 : List

Ways to use `markovMatrices` :

markovMatrices(Ring,Digraph,List)

markovMatrices -- The matrices whose minors form the ideal associated to the list of independence statements of the graph.

Synopsis

Description

See also

Ways to use markovMatrices :

Ways to use `markovMatrices` :