Matrix completion theorems

Newman, Morris

doi:10.1090/S0002-9939-1985-0781052-8

Matrix completion theorems
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by Morris Newman PDF

Proc. Amer. Math. Soc. 94 (1985), 39-45 Request permission

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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