From: Robert Bryant <bryant@math.duke.edu>
Subject: Re: minors of a singular matrix
Date: 20 Feb 1999 15:30:08 -0600
Newsgroups: sci.math.research

Dima Pasechnik wrote:
 
> Let X be an n x n matrix such that u^T X = X v = 0.
> Define n x n matrix A as follows:
> (A_{ij}) = (-1)^{i+j} |M_{ij}|,
> where M_{ij} is the matrix obtained from X by removal of
> i-th row and j-th column.
> 
> Then A=uv^T.               (*)
> 
> The questions:
> 
> 1) Does the relation (*) have a name, and is it in some book?
> 
> 2) Can (*) be generalized for (n-k)x(n-k) minors of X, in
> particular when rank X = n-k ?

  Well, I wouldn't say it has a name exactly, but there
is a conceptual interpretation of this result that lends
itself very well to the generalization to the (n-k) x (n-k)
minors case.  It involves exterior algebra:

	If V and W are vector spaces and X:V -> W is a 
linear map, then X  canonically induces a algebra
homomorphism  E(X):E(V) -> E(W), where E(V) is the 
exterior algebra generated by V.  The algebra E(V)
is graded, with E_0(V) = F (the ground field, in your
case, the reals) and E_1(V) = V.  If V has dimension n
over F, then E_p(V)= 0 for p > n and, in general 
E_p(V) has dimension (n \choose p)  when 0 \le p \le n.
Once you choose bases of V and W, so that X can
be regarded as a matrix, the spaces E_k(V) and E_k(W)
have canonically associated bases and the linear
map  E_k(X): E_k(V) -> E_k(W)  has the k x k minors
of the matrix of X as its matrix entries.  
	If K is the kernel of X and I is the image of X,
then X induces an isomorphism Y: V/K -> I and 
consequently an isomorphism  E(Y): E(V/K) -> E(I)
that makes the diagram below commute:

     E(V)  --->  E(W)

      |           ^     (excuse the clumsy diagram)
      v           |

     E(V/K) -->  E(I)

In particular, it follows that the rank of E_k(X)
is the same as the rank of E_k(Y), which is clearly
(r \choose k), where r is the rank of X.  In particular,
the rank of the matrix of E_r(X), i.e., the matrix
of r x r minors of X, is 1, and hence can be factored
as a column vector times a row vector.  Moreover, the
column and row vectors are determined (up to scalar
multiples) by the kernel K in V and the image I in W.
	Your case (with the scalar multiplier inserted, as 
Robert Israel correctly pointed out)  is an application 
of this to the case where r = n-1  (Of course, if
the rank of X is less than n-1, E_{n-1}(X) will 
be zero.)

	Yours,

	Robert Bryant