From: Robert Bryant
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