Corpus ID: 219177107

Effectively bounded idempotent generation of certain $2 \times 2$ singular matrices by idempotent matrices over real quadratic number rings

@article{Nguyen2020EffectivelyBI,
  title={Effectively bounded idempotent generation of certain \$2 \times 2\$ singular matrices by idempotent matrices over real quadratic number rings},
  author={D. Q. Nguyen},
  journal={arXiv: Number Theory},
  year={2020}
}
  • D. Q. Nguyen
  • Published 2020
  • Mathematics
  • arXiv: Number Theory
  • Let $k = \mathbb{Q}(\sqrt{\alpha})$ be a real quadratic number field, where $\alpha$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular… CONTINUE READING

    References

    Publications referenced by this paper.
    SHOWING 1-10 OF 12 REFERENCES
    microcredit and rural finance. Beijing: China
    • 2006
    The ALEP Definition of Complementarity and Least Concave Utility Functions
    40
    Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings
    2
    Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf
    • 1966
    Bounded generation of SL2 over rings of S-integers with infinitely many units, Algebra
    • 2018
    Factorizations of integer matrices as products of idempotents and nilpotents
    5
    Products of idempotent matrices, Linear and Multilinear Algebra
    • 1983
    Bounded generation of SL 2 over rings of S-integers with infinitely many units
    • 2018
    Factorization of singular integer matrices
    5