Two theorems on integral matrices

Abstract

The following two results are proved: (1) For a positive definite integral symmetric matrix S of rank (S) < 7 or when rank (S) = 8, S has an odd entry in its diagonal, there is an integral matrix A satisfying AA^t = Sif there is a rational matrix R with RR^t = S (2) Given an integral matrix A of size r×n such that AA^t = mI_r there is then always an integral completion matrix B of size n×n satisfying BB^t = mI_r whenever n-r is less than or equal to 7. This threshold number 7 is the best possible. (Here m, n satisfy the obvious necessary conditions.)

^†Research supported in part by National Science Foundation.

^†Research supported in part by National Science Foundation.

Notes

^†Research supported in part by National Science Foundation.