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allUniquePrincipalMinors -- Compute all unique principal minors of a given matrix

Description

The function allUniquePrincipalMinors computes all unique principal minors of a given square matrix. A principal minor of a matrix is the determinant of a submatrix obtained by deleting the same set of rows and columns from the matrix. The function returns a list of all unique principal minors of the given matrix.

i1 : G = graph({0,1,2,3},{{0,1},{1,2},{2,3},{0,3}})

o1 = Graph{0 => {1, 3}}
           1 => {0, 2}
           2 => {1, 3}
           3 => {0, 2}

o1 : Graph
i2 : S = oscRing(G, CoefficientRing => QQ, Reduced => true)

o2 = S

o2 : PolynomialRing
i3 : I = oscSystem(G,S);

o3 : Ideal of S
i4 : C = decompose I;
i5 : J = oscJacobian(G,S)

o5 = | -x_1-x_3 x_1                0                           
     | x_1      -x_1x_2-y_1y_2-x_1 x_1x_2+y_1y_2               
     | 0        x_1x_2+y_1y_2      -x_1x_2-x_2x_3-y_1y_2-y_2y_3
     | x_3      0                  x_2x_3+y_2y_3               
     ------------------------------------------------------------------------
     x_3                |
     0                  |
     x_2x_3+y_2y_3      |
     -x_2x_3-y_2y_3-x_3 |

             4      4
o5 : Matrix S  <-- S
i6 : netList for i in C list allUniquePrincipalMinors(-J, Modulo=>i)

     +---+------+---------+----+---------+
     |   |      | 2       |    |  2      |
o6 = |2x |0     |y  - 1   |-2x |4y  - 4  |
     |  3|      | 3       |   3|  3      |
     +---+------+---------+----+---------+
     |   | 2    |         |    |         |
     |0  |y  - 1|         |    |         |
     |   | 3    |         |    |         |
     +---+------+---------+----+---------+
     |   |      |  1     1|    |         |
     |0  |-2x   |- -x  - -|2x  |- 2x  - 2|
     |   |   3  |  2 2   2|  3 |    2    |
     +---+------+---------+----+---------+
     |2  |3     |4        |0   |         |
     +---+------+---------+----+---------+
     |-2 |3     |4        |-4  |0        |
     +---+------+---------+----+---------+

See also

Ways to use allUniquePrincipalMinors:

  • allUniquePrincipalMinors(Matrix)

For the programmer

The object allUniquePrincipalMinors is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:645:0.