Re: minors and cofactors

[johan19]

On Fri, Sep 18, 1998 at 06:44:39PM +0200, Dirk Laurie wrote:
> Thanks to David Clark for a fascinating question.  I keep getting
> new thoughts after having posted two answers, but this is the last
> one.
> 
> There is a fully satisfactory way of calculating the cofactor matrix
> that requires no rank decision.
> 
> Calculate the SVD  A=U*S*V'.
> Then det(A)=det(U)*det(S)*det(V')
> and  inv(A)=V*inv(S)*U'.
> Hence  inv(A)*det(A) = (det(V')*V) * (det(S)*inv(S)) * (det(U)*U')
> The middle factor can be calculated by replacing each diagonal term
> be the product of the others.

just to throw my two cents in...

note that U and V in the SVD are *unitary* matrices and hence have
det(U)=det(V)=1.  S is diagonal hence the determinant is trivial to
compute.

simpleton's (that'd be me) wild-ass guess, is the cofactor related to
deleting a row from both U and V and forming the product Uk*S*Vk' or
Vk*S*Uk' where Uk and Vk are U and V missing row k?  

(i'd check myself, but i haven't been able to recompile octave 2.1.7
since upgrading to egcs-1.1 and the new libstdc++ isn't compatible.
unfortunately, i blew away my old libstdc++ so i guess it may be a day
or two before i get a working one again...)

--
Johan Kullstam address@hidden Don't Fear the Penguin!