axiom-developer
[Top][All Lists]

## [Axiom-developer] Re: pfaffian.input.pamphlet

 From: root Subject: [Axiom-developer] Re: pfaffian.input.pamphlet Date: Sat, 29 Sep 2007 22:32:24 -0400

```>> 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.

You're quite correct. It seems that the documentation on wikipedia
is wrong, at least as I read it. Do you agree?

Tim

```

reply via email to