adjointFactorization -- Constructs matrix factorization of a determinant of a matrix

Usage:

adjointFactorization M
Inputs:
- M, a matrix, A square matrix
Outputs:
- a ZZ/d graded factorization, The matrix factorization of the determinant of M determined by the adjoint matrix

Description

Given a matrix $M$, the adjoint matrix $M'$ is the matrix satisfying $M \cdot M' = M' \cdot M = \det(M) id$. This means that any square matrix induces a matrix factorization of its determinant, and this function implements this induced matrix factorization.

i1 : Q = QQ[x_(1,1)..x_(3,3)]

o1 = Q

o1 : PolynomialRing

i2 : M = genericMatrix(Q,3,3)

o2 = | x_(1,1) x_(2,1) x_(3,1) |
     | x_(1,2) x_(2,2) x_(3,2) |
     | x_(1,3) x_(2,3) x_(3,3) |

             3      3
o2 : Matrix Q  <-- Q

i3 : F = adjointFactorization M

      3      3      3
o3 = Q  <-- Q  <-- Q
                    
     0      1      0

o3 : ZZdFactorization

i4 : F.dd

          3                                                                                                            3
o4 = 1 : Q  <-------------------------------------------------------------------------------------------------------- Q  : 0
               {1} | -x_(2,3)x_(3,2)+x_(2,2)x_(3,3) x_(2,3)x_(3,1)-x_(2,1)x_(3,3)  -x_(2,2)x_(3,1)+x_(2,1)x_(3,2) |
               {1} | x_(1,3)x_(3,2)-x_(1,2)x_(3,3)  -x_(1,3)x_(3,1)+x_(1,1)x_(3,3) x_(1,2)x_(3,1)-x_(1,1)x_(3,2)  |
               {1} | -x_(1,3)x_(2,2)+x_(1,2)x_(2,3) x_(1,3)x_(2,1)-x_(1,1)x_(2,3)  -x_(1,2)x_(2,1)+x_(1,1)x_(2,2) |

          3                                       3
     0 : Q  <----------------------------------- Q  : 1
               {2} | x_(1,1) x_(2,1) x_(3,1) |
               {2} | x_(1,2) x_(2,2) x_(3,2) |
               {2} | x_(1,3) x_(2,3) x_(3,3) |

o4 : ZZdFactorizationMap

i5 : potential F == determinant M

o5 = true

Ways to use adjointFactorization:

adjointFactorization(Matrix)

For the programmer

The object adjointFactorization is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:6350:0.

adjointFactorization -- Constructs matrix factorization of a determinant of a matrix

Description

See also

Ways to use adjointFactorization:

For the programmer