---
_id: '14780'
abstract:
- lang: eng
  text: In this paper, we study the eigenvalues and eigenvectors of the spiked invariant
    multiplicative models when the randomness is from Haar matrices. We establish
    the limits of the outlier eigenvalues λˆi and the generalized components (⟨v,uˆi⟩
    for any deterministic vector v) of the outlier eigenvectors uˆi with optimal convergence
    rates. Moreover, we prove that the non-outlier eigenvalues stick with those of
    the unspiked matrices and the non-outlier eigenvectors are delocalized. The results
    also hold near the so-called BBP transition and for degenerate spikes. On one
    hand, our results can be regarded as a refinement of the counterparts of [12]
    under additional regularity conditions. On the other hand, they can be viewed
    as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar
    random matrix.
acknowledgement: The authors would like to thank the editor, the associated editor
  and two anonymous referees for their many critical suggestions which have significantly
  improved the paper. The authors are also grateful to Zhigang Bao and Ji Oon Lee
  for many helpful discussions. The first author also wants to thank Hari Bercovici
  for many useful comments. The first author is partially supported by National Science
  Foundation DMS-2113489 and the second author is supported by ERC Advanced Grant
  “RMTBeyond” No. 101020331.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Xiucai
  full_name: Ding, Xiucai
  last_name: Ding
- first_name: Hong Chang
  full_name: Ji, Hong Chang
  id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d
  last_name: Ji
citation:
  ama: Ding X, Ji HC. Spiked multiplicative random matrices and principal components.
    <i>Stochastic Processes and their Applications</i>. 2023;163:25-60. doi:<a href="https://doi.org/10.1016/j.spa.2023.05.009">10.1016/j.spa.2023.05.009</a>
  apa: Ding, X., &#38; Ji, H. C. (2023). Spiked multiplicative random matrices and
    principal components. <i>Stochastic Processes and Their Applications</i>. Elsevier.
    <a href="https://doi.org/10.1016/j.spa.2023.05.009">https://doi.org/10.1016/j.spa.2023.05.009</a>
  chicago: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices
    and Principal Components.” <i>Stochastic Processes and Their Applications</i>.
    Elsevier, 2023. <a href="https://doi.org/10.1016/j.spa.2023.05.009">https://doi.org/10.1016/j.spa.2023.05.009</a>.
  ieee: X. Ding and H. C. Ji, “Spiked multiplicative random matrices and principal
    components,” <i>Stochastic Processes and their Applications</i>, vol. 163. Elsevier,
    pp. 25–60, 2023.
  ista: Ding X, Ji HC. 2023. Spiked multiplicative random matrices and principal components.
    Stochastic Processes and their Applications. 163, 25–60.
  mla: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices and
    Principal Components.” <i>Stochastic Processes and Their Applications</i>, vol.
    163, Elsevier, 2023, pp. 25–60, doi:<a href="https://doi.org/10.1016/j.spa.2023.05.009">10.1016/j.spa.2023.05.009</a>.
  short: X. Ding, H.C. Ji, Stochastic Processes and Their Applications 163 (2023)
    25–60.
date_created: 2024-01-10T09:29:25Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2024-01-16T08:49:51Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1016/j.spa.2023.05.009
ec_funded: 1
external_id:
  arxiv:
  - '2302.13502'
  isi:
  - '001113615900001'
file:
- access_level: open_access
  checksum: 46a708b0cd5569a73d0f3d6c3e0a44dc
  content_type: application/pdf
  creator: dernst
  date_created: 2024-01-16T08:47:31Z
  date_updated: 2024-01-16T08:47:31Z
  file_id: '14806'
  file_name: 2023_StochasticProcAppl_Ding.pdf
  file_size: 1870349
  relation: main_file
  success: 1
file_date_updated: 2024-01-16T08:47:31Z
has_accepted_license: '1'
intvolume: '       163'
isi: 1
keyword:
- Applied Mathematics
- Modeling and Simulation
- Statistics and Probability
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 25-60
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Stochastic Processes and their Applications
publication_identifier:
  eissn:
  - 1879-209X
  issn:
  - 0304-4149
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: Spiked multiplicative random matrices and principal components
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 163
year: '2023'
...