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Proceedings of the American Mathematical Society

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

 

 

Matrix completion theorems


Author: Morris Newman
Journal: Proc. Amer. Math. Soc. 94 (1985), 39-45
MSC: Primary 15A33; Secondary 15A57
DOI: https://doi.org/10.1090/S0002-9939-1985-0781052-8
MathSciNet review: 781052
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Abstract: Let $ R$ be a principal ideal ring, $ {M_{t,n}}$ the set of $ t \times n$ matrices over $ R$. The following results are proved:

(a) Let $ D \in {M_{n,n}}$. Then the least nonnegative integer $ t$ such that a matrix $ \left[ {\begin{array}{*{20}{c}} * & * \\ * & D \\ \end{array} } \right]$ exists which belongs to $ {\text{GL(}}n + t,R)$ is $ t = n - p$, where $ p$ is the number of invariant factors of $ D$ equal to 1.

(b) Any primitive element of $ {M_{1,2n}}$ may be completed to a $ 2n \times 2n$ symplectic matrix.

(c) If $ A,B \in {M_{n,n}}$ are such that $ [A,B]$ is primitive and $ A{B^T}$ is symmetric, then $ [A,B]$ may be completed to a $ 2n \times 2n$ symplectic matrix.

(d) If $ A \in {M_{t,t}},B \in {M_{t,n - t}}$, are such that $ [A,B]$ is primitive and $ A$ is symmetric, then $ [A,B]$ may be completed to a symmetric element of $ {\text{SL(}}n,R{\text{)}}$, provided that $ 1 \leqslant t \leqslant n/3$.

(e) If $ n \geqslant 3$, then any primitive element of $ {M_{1,n}}$ occurs as the first row of the commutator of two elements of $ {\text{SL(}}n,R{\text{)}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0781052-8
Article copyright: © Copyright 1985 American Mathematical Society