---
_id: '12179'
abstract:
- lang: eng
  text: We derive an accurate lower tail estimate on the lowest singular value σ1(X−z)
    of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z.
    Such shift effectively changes the upper tail behavior of the condition number
    κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices
    to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away
    from the real axis. This sharpens and resolves a recent conjecture in [J. Banks
    et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of
    the real Ginibre ensemble with a genuinely complex shift. As a consequence we
    obtain an improved upper bound on the eigenvalue condition numbers (known also
    as the eigenvector overlaps) for real Ginibre matrices. The main technical tool
    is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys.,
    1 (2020), pp. 101--146].
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Dominik J
  full_name: Schröder, Dominik J
  id: 408ED176-F248-11E8-B48F-1D18A9856A87
  last_name: Schröder
  orcid: 0000-0002-2904-1856
citation:
  ama: Cipolloni G, Erdös L, Schröder DJ. On the condition number of the shifted real
    Ginibre ensemble. <i>SIAM Journal on Matrix Analysis and Applications</i>. 2022;43(3):1469-1487.
    doi:<a href="https://doi.org/10.1137/21m1424408">10.1137/21m1424408</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). On the condition number
    of the shifted real Ginibre ensemble. <i>SIAM Journal on Matrix Analysis and Applications</i>.
    Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/21m1424408">https://doi.org/10.1137/21m1424408</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “On the Condition
    Number of the Shifted Real Ginibre Ensemble.” <i>SIAM Journal on Matrix Analysis
    and Applications</i>. Society for Industrial and Applied Mathematics, 2022. <a
    href="https://doi.org/10.1137/21m1424408">https://doi.org/10.1137/21m1424408</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “On the condition number of the
    shifted real Ginibre ensemble,” <i>SIAM Journal on Matrix Analysis and Applications</i>,
    vol. 43, no. 3. Society for Industrial and Applied Mathematics, pp. 1469–1487,
    2022.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2022. On the condition number of the shifted
    real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. 43(3),
    1469–1487.
  mla: Cipolloni, Giorgio, et al. “On the Condition Number of the Shifted Real Ginibre
    Ensemble.” <i>SIAM Journal on Matrix Analysis and Applications</i>, vol. 43, no.
    3, Society for Industrial and Applied Mathematics, 2022, pp. 1469–87, doi:<a href="https://doi.org/10.1137/21m1424408">10.1137/21m1424408</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, SIAM Journal on Matrix Analysis and
    Applications 43 (2022) 1469–1487.
date_created: 2023-01-12T12:12:38Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2023-01-27T06:56:06Z
day: '01'
department:
- _id: LaEr
doi: 10.1137/21m1424408
external_id:
  arxiv:
  - '2105.13719'
intvolume: '        43'
issue: '3'
keyword:
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2105.13719
month: '07'
oa: 1
oa_version: Preprint
page: 1469-1487
publication: SIAM Journal on Matrix Analysis and Applications
publication_identifier:
  eissn:
  - 1095-7162
  issn:
  - 0895-4798
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the condition number of the shifted real Ginibre ensemble
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2022'
...