berezinian -- Computes the berezinian of a supermatrix.

Synopsis

Usage:

N = berezinian(G, R)
Inputs:
- G, an instance of the type SuperMatrix,
- R, a ring,
Outputs:
- N, a number,

Description

This function works only when the entries of the even blocks are numbers, and those of odd blocks are formed by odd generators. If in a super Matrix, one of the first or the second diagonal blocks is invertible, then we can define the berezinian (as a kind of super Determinant). The formula for the berezinian is different base on which block is invertible. But it is shown that the two formulas are equivalent if two blocks are invertible. If $M=\begin{pmatrix} M1&M2\\ M3&M4\end{pmatrix}$. is a super Matrix, and $M_4$ is invertible, then $Ber(M)= det(M_1-M_2M^{-1}_4M_3) det(M_4)^{-1}$.

If $M_1$ is invertible, then $Ber(M) = det(M_4-M_3M_1^{-1}M_2)^{-1} det(M_1)$.

i1 : M1 = matrix{{5, 7}, {1, 2}}

o1 = | 5 7 |
     | 1 2 |

              2       2
o1 : Matrix ZZ  <-- ZZ

i2 : M2 = matrix{{1, 2, 3}, {4, 5, 6}}

o2 = | 1 2 3 |
     | 4 5 6 |

              2       3
o2 : Matrix ZZ  <-- ZZ

i3 : M3 = matrix{{3, 4}, {5, 6}, {7, 8}}

o3 = | 3 4 |
     | 5 6 |
     | 7 8 |

              3       2
o3 : Matrix ZZ  <-- ZZ

i4 : M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}}

o4 = | 2 3 11 |
     | 4 5 6  |
     | 7 8 9  |

              3       3
o4 : Matrix ZZ  <-- ZZ

i5 : M5 = sub(M4, QQ)

o5 = | 2 3 11 |
     | 4 5 6  |
     | 7 8 9  |

              3       3
o5 : Matrix QQ  <-- QQ

i6 : G = superMatrixGenerator(M1, M2, M3, M4)

o6 = SuperMatrix{...1...}

o6 : SuperMatrix

i7 : berezinian(G, QQ)

        4
o7 = - --
       21

o7 : QQ

berezinian -- Computes the berezinian of a supermatrix.

Synopsis

Description

Caveat

Ways to use berezinian :

For the programmer