Symmetric Completions and Products of Symmetric Matrices
Transactions of the American Mathematical Society
We show that any vector of n relatively prime coordinates from a principal ideal ring R may be completed to a symmetric matrix of SL(n, R), provided that n a 4. The result is also true for n = 3 if R is the ring of integers Z. This implies for example that if F is a field, any matrix of SL(n, F) is the product of a fixed number of symmetric matrices of SL(r¡, F) except when n = 2, F = GF(3), which is a genuine exception. Introduction-. It is known that any matrix over a field may be expressed
... may be expressed as the product of two symmetric matrices over that field (see [l],  ). This classical result has been taken up again by 0. Taussky (see , ) who considered the problem of factoring an integral matrix into the product of two integral symmetric matrices, and showed among other things that such a factorization is not always possible; for example the matrix [~58 g] is not the product of two integral symmetric matrices. An interesting problem is to determine whether some result of this kind remains true when the matrices are restricted to lie in SL(n, R), where R is a principal ideal ring. We prove for example that if F is a field, then any matrix of SL(n, F) is the product of a fixed number of symmetric matrices of SL(n, F), except when n = 2, F = GF(3). This case is a genuine exception: there are matrices of SL(2, GF (3) ) which are not expressible as the product of finitely many symmetric matrices of SL(2, GF (3) ). This paper is in two sections. In the first, we derive a number of results on completing rectangular arrays over R to unimodular symmetric matrices over R. In the second, we use these results to derive theorems of the type quoted above on factorization of matrices into products of symmetric matrices. The results of the first section are analogous to the classical ones on completion of rectangular arrays to unimodular matrices, but are naturally more difficult because of the added requirement of symmetry. These results should be useful in other applications. The results of the second section suggest some interesting open problems, which are discussed at the end of the paper.