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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An answer to a question of M. Newman on matrix completion


Author: L. N. Vaserstein
Journal: Proc. Amer. Math. Soc. 97 (1986), 189-196
MSC: Primary 18F25; Secondary 13D15, 15A33, 19B10
DOI: https://doi.org/10.1090/S0002-9939-1986-0835863-1
MathSciNet review: 835863
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Abstract: Let $ R$ be a principal ideal ring, $ A$ a symmetric $ t$-by-$ t$ matrix over $ R$, $ B$ a $ t$-by-$ (n - t)$ matrix over $ R$ such that the $ t$-by-$ n$ matrix $ (A,B)$ is primitive. Newman [2] proved that $ (A,B)$ may be completed (as the first $ t$ rows) to a symmetric $ n$-by-$ n$ matrix of determinant 1, provided that $ 1 \leq t \leq n/3$. He showed that the result is false, in general, if $ t = n/2$, and he asked to determine all values of $ t$ such that $ 1 \leq t \leq n$ and the result holds. It is shown here that these values are exactly $ t$ satisfying $ 1 \leq t \leq n/2$.

Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass [1].

Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].


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DOI: https://doi.org/10.1090/S0002-9939-1986-0835863-1
Article copyright: © Copyright 1986 American Mathematical Society