[Axiom-math] Re: pfaffian.input.pamphlet

[Martin Rubey]

root <address@hidden> writes:

> Martin,
> 
> I've been studying Pfaffian systems (related to robotics, a field I
> know a bit about). It seems that your algorithm does not work for
> embedded matrices as in:
> 
> z:SQMATRIX(2,INT):=[[0,0],[0,0]]
> m:SQMATRIX(2,INT):=[[0,1],[-1,0]]
> m2:SQMATRIX(4,SQMATRIX(2,INT)):=[[m,z,z,z],[z,m,z,z],[z,z,m,z],[z,z,z,m]]
> which should be 1*1*1*1 = 1
> 
> Am I doing something wrong?

The matrix m2 is not skew-symmetric: it should be zero along the diagonal, but
it contains m along the diagonal.  Try the following instead:

(19) -> z:SQMATRIX(2,INT):=[[0,0],[0,0]]; m:SQMATRIX(2,INT):=[[0,1],[-1,0]];
        m2:= matrix [[z,m,m,m],[-m,z,z,z],[-m,z,z,m],[-m,z,-m,z]]

         + +0  0+   + 0   1+  + 0   1+  + 0   1++
         | |    |   |      |  |      |  |      ||
         | +0  0+   +- 1  0+  +- 1  0+  +- 1  0+|
         |                                      |
         |+0  - 1+   +0  0+    +0  0+    +0  0+ |
         ||      |   |    |    |    |    |    | |
         |+1   0 +   +0  0+    +0  0+    +0  0+ |
   (19)  |                                      |
         |+0  - 1+   +0  0+    +0  0+   + 0   1+|
         ||      |   |    |    |    |   |      ||
         |+1   0 +   +0  0+    +0  0+   +- 1  0+|
         |                                      |
         |+0  - 1+   +0  0+   +0  - 1+   +0  0+ |
         ||      |   |    |   |      |   |    | |
         ++1   0 +   +0  0+   +1   0 +   +0  0+ +
                                         Type: Matrix SquareMatrix(2,Integer)
(20) -> PfChar(l, m2)

          4   + 0   2+ 2   +- 1   0 +
   (20)  l  + |      |l  + |        |
              +- 2  0+     + 0   - 1+
                                     Type: Polynomial SquareMatrix(2,Integer)

So, the Pfaffian is

+- 1   0 +
|        |
+ 0   - 1+

In particular, look at its type: the Pfaffiam is an element of the groung ring.

Martin