An answer to a question of M. Newman on matrix completion
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- by L. N. Vaserstein PDF
- Proc. Amer. Math. Soc. 97 (1986), 189-196 Request permission
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].References
- H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
- Morris Newman, Matrix completion theorems, Proc. Amer. Math. Soc. 94 (1985), no. 1, 39–45. MR 781052, DOI 10.1090/S0002-9939-1985-0781052-8
- L. N. Vaseršteĭn, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 17–27 (Russian). MR 0284476
- L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81 (123) (1970), 328–351 (Russian). MR 0269722
- L. N. Vaseršteĭn, Stabilization for the classical groups over rings, Mat. Sb. (N.S.) 93(135) (1974), 268–295, 327 (Russian). MR 0338208
- L. N. Vaserstein, Bass’s first stable range condition, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 319–330. MR 772066, DOI 10.1016/0022-4049(84)90044-6
- L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
Additional Information
- © Copyright 1986 American Mathematical Society
- 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