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 ≦ t ≦ 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 ≦ t ≦ n and the result holds. It is shown here that these values are exactly t satisfying 1 ≦ t < 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].
| Original language | English (US) |
|---|---|
| Pages (from-to) | 189-196 |
| Number of pages | 8 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 97 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jan 1 1986 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
- Applied Mathematics
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An answer to a question of M. Newman on matrix completion. / Vaserstein, Leonid N.
In: Proceedings of the American Mathematical Society, Vol. 97, No. 2, 01.01.1986, p. 189-196.Research output: Contribution to journal › Article
TY - JOUR
T1 - An answer to a question of M. Newman on matrix completion
AU - Vaserstein, Leonid N.
PY - 1986/1/1
Y1 - 1986/1/1
N2 - 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 ≦ t ≦ 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 ≦ t ≦ n and the result holds. It is shown here that these values are exactly t satisfying 1 ≦ t < 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].
AB - 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 ≦ t ≦ 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 ≦ t ≦ n and the result holds. It is shown here that these values are exactly t satisfying 1 ≦ t < 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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U2 - 10.1090/S0002-9939-1986-0835863-1
DO - 10.1090/S0002-9939-1986-0835863-1
M3 - Article
AN - SCOPUS:84968482104
VL - 97
SP - 189
EP - 196
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 2
ER -