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[Axiom-math] Re: pfaffian.input.pamphlet
From: |
root |
Subject: |
[Axiom-math] 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