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