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

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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  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 . 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) 189-196 8 Proceedings of the American Mathematical Society 97 2 https://doi.org/10.1090/S0002-9939-1986-0835863-1 Published - Jun 1986

### Fingerprint

Matrix Completion
Ring
Commutative Ring
Determinant
Theorem
Range of data

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics

### Cite this

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title = "An answer to a question of M. Newman on matrix completion",
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  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 . 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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}

In: Proceedings of the American Mathematical Society, Vol. 97, No. 2, 06.1986, p. 189-196.

Research output: Contribution to journalArticle

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T1 - An answer to a question of M. Newman on matrix completion

AU - Vaserstein, L. N.

PY - 1986/6

Y1 - 1986/6

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  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 . 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  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 . 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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