Skip to content
DraconiticResearch
Search Sources API
189,273 studies31 sourcesUpdated 48 min ago
PreprintarXiv licencearXiv

Towards a Characterization of Matrix Monotonicity via Radial Convexity

  • Małgorzata M. Czerwińska
  • Jose A. Franco
  • Allan Merino
Open at sourcePDFDOICite

Abstract

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order n is equivalent to matrix monotonicity of degree n, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

Related

  1. Convexity of a ratio function involving polygamma functions and monotonicity of the sectional curvature of the Fisher--Rao manifold
    Feng Qi
    arXivPreprintCC BY 4.0Full textGeneral mathematicsarXiv
  2. Vector and matrix monotonicity characterisation of the "load forces" in nonlinear elasticity: compressible versus incompressible response
    Jendrik Voss, Ionel-Dumitrel Ghiba, Robert J. Martin +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. Quantum Matrix-Product Codes: CSS-T Characterization and Maximality
    Delio Jaramillo-Velez, Alessandro Neri, Adway Patra +2
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  4. A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching
    Chaochao Zhu, Qin Yue
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Support monotonicity and gap dynamics in one-dimensional MMD flows
    Karim Bounja, Abdeljalil Sakat
    HALPreprintPublisher termsFull textAnalysis & PDEsOptimisation & controlHAL

Cite

BibTeXRIS
BibTeX
@misc{czerwinska2026towards,
  title = {{Towards a Characterization of Matrix Monotonicity via Radial Convexity}},
  author = {Małgorzata M. Czerwińska and Jose A. Franco and Allan Merino},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17393},
  eprint = {2610.17393},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17393},
}
RIS
TY  - UNPB
TI  - Towards a Characterization of Matrix Monotonicity via Radial Convexity
AU  - Małgorzata M. Czerwińska
AU  - Jose A. Franco
AU  - Allan Merino
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17393
UR  - https://arxiv.org/abs/2610.17393
AB  - The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order n is equivalent to matrix monotonicity of degree n, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.
ER  -