A Complementarity Property of Rank-One Multiplicative Maps

Abstract

The semidefinite linear complementarity problem associated with the multiplicative map M_A:𝕊ⁿ→𝕊ⁿ defined by M_A(X)=AXA^T, is considered, in the case when A∈ℝ^{n× n} has rank one. A linear map is said to have the Q₀-property if every feasible instance of the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When A=xy^T with x,y∈ℝⁿ \emph{linearly independent}, we construct a matrix Q∈𝕊ⁿ for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that M_A \emph{does not have} the Q₀-property. Conversely, when A=uu^T for some nonzero u∈ℝⁿ, we prove that M_A \emph{has} the Q₀-property. These two results provide a complete characterization of the Q₀-property within the family of rank-one multiplicative maps.

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BibTeX RIS

BibTeX

@misc{sivakumar2026complementarity,
  title = {{A Complementarity Property of Rank-One Multiplicative Maps}},
  author = {K. C. Sivakumar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37197},
  eprint = {2609.37197},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37197},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - A Complementarity Property of Rank-One Multiplicative Maps
AU  - K. C. Sivakumar
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37197
UR  - https://arxiv.org/abs/2609.37197
AB  - The semidefinite linear complementarity problem associated with the multiplicative map M_A:𝕊^n→𝕊^n defined by M_A(X)=AXA^T, is considered, in the case when A∈ℝ^n× n has rank one. A linear map is said to have the Q_0-property if every feasible instance of the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When A=xy^T with x,y∈ℝ^n \emph{linearly independent}, we construct a matrix Q∈𝕊^n for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that M_A \emph{does not have} the Q_0-property. Conversely, when A=uu^T for some nonzero u∈ℝ^n, we prove that M_A \emph{has} the Q_0-property. These two results provide a complete characterization of the Q_0-property within the family of rank-one multiplicative maps.
ER  -

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