Symmetric completions and products of symmetric matrices

Abstract:

We show that any vector of n relatively prime coordinates from a principal ideal ring R may be completed to a symmetric matrix of ${\text {SL}}(n,R)$, provided that $n \geq 4$. 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 ${\text {SL}}(n,F)$ is the product of a fixed number of symmetric matrices of ${\text {SL}}(n,F)$ except when $n = 2$, $F = {\text {GF}}(3)$, which is a genuine exception.