Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946.
  • [2] M. Newman, Integral matrices, Academic Press, New York, 1972. MR 0340283 (49:5038)
  • [3] O. Taussky, The role of symmetric matrices in the study of general matrices, Linear Algebra and Appl. 5 (1972), 147-154. MR 0302674 (46:1818)
  • [4] -, The factorization of an integral matrix into a product of two integral symmetric matrices. I, Acta Arith. (to appear). MR 0335551 (49:332)
  • [5] -, The factorization of an integral matrix into a product of two integral symmetric matrices. II, Comm. Pure Appl. Math. (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A33

Retrieve articles in all journals with MSC: 15A33


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

American Mathematical Society