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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Symmetric completions and products of symmetric matrices


Author: Morris Newman
Journal: Trans. Amer. Math. Soc. 186 (1973), 191-201
MSC: Primary 15A33
DOI: https://doi.org/10.1090/S0002-9947-1973-0485931-7
MathSciNet review: 0485931
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0485931-7
Keywords: Principal ideal rings, fields, symmetric matrices, unimodular matrices, symmetric completion
Article copyright: © Copyright 1973 American Mathematical Society