This appendix presents the theory of invariant factors (also called elementary divisors) of an integral matrix, which we require in §2.3. For convenience, we also extend the notion of invariant factors to rational matrices.

Theorem C.I.Let A be an m x m matrix withintegral entries. There exist integral unimodular matrices P, Q such that

(1) PAQ is a diagonal matrix, PAQ=diag(d,…,dm)

(2) didivides di+1, for i=l,…, m-1.

Moreover the diare determined up to sign and are called the invariant factors of A.

Proof. If B and C are integral m x m matrices, write B C if C can be obtained from B by a finite sequence of elementary row and column operations; that is, if C=PBQ where P and Q are products of elementary matrices (and thus, in particular, unimodular). Also, if B is a nonzero matrix, let ø (B) be the minimum of the absolute values of the nonzero entries of B.

If A=0 there is clearly nothing to prove. So, assume that A≠O. Choose B such that and ø(B) is minimal. After possibly permuting rows and columns we may assume that b11, is a nonzero entry of minimal absolute value in B.