Matrix completion theorems
HTML articles powered by AMS MathViewer
 by Morris Newman PDF
 Proc. Amer. Math. Soc. 94 (1985), 3945 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 ${\text {SL(}}n,R{\text {)}}$.References

C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946.
 Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New YorkLondon, 1972. MR 0340283
 Morris Newman, Symmetric completions and products of symmetric matrices, Trans. Amer. Math. Soc. 186 (1973), 191–210 (1974). MR 485931, DOI 10.1090/S00029947197304859317
 Irving Reiner, Symplectic modular complements, Trans. Amer. Math. Soc. 77 (1954), 498–505. MR 67076, DOI 10.1090/S00029947195400670761
 Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527–606 (German). MR 1503238, DOI 10.2307/1968644
Additional Information
 © Copyright 1985 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 94 (1985), 3945
 MSC: Primary 15A33; Secondary 15A57
 DOI: https://doi.org/10.1090/S00029939198507810528
 MathSciNet review: 781052