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 , provided that . The result is also true for if *R* is the ring of integers *Z*. This implies for example that if *F* is a field, any matrix of is the product of a fixed number of symmetric matrices of except when , , which is a genuine exception.

**[1]**C. C. MacDuffee,*The theory of matrices*, Chelsea, New York, 1946.**[2]**Morris Newman,*Integral matrices*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR**0340283****[3]**Olga Taussky,*The role of symmetric matrices in the study of general matrices*, Linear Algebra and Appl.**5**(1972), 147–154. MR**0302674****[4]**Olga Taussky,*The factorization of an integral matrix into a product of two integral symmetric matrices. I*, Acta Arith.**24**(1973), 151–156. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, II. MR**0335551**, https://doi.org/10.4064/aa-24-2-151-156**[5]**-,*The factorization of an integral matrix into a product of two integral symmetric matrices*. II, Comm. Pure Appl. Math. (to appear).

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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