Macaulay2 » Documentation
Packages » MatrixFactorizations :: adjointFactorization
next | previous | forward | backward | up | index | toc

adjointFactorization -- Constructs matrix factorization of a determinant of a 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

See also

Ways to use adjointFactorization:

  • adjointFactorization(Matrix)

For the programmer

The object adjointFactorization is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:6350:0.