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Monotone transformations on the cone of all positive semidefinite real matrices

  • Iva Golubić EMAIL logo and Janko Marovt
Published/Copyright: May 23, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 3

Abstract

Let Hn+(ℝ) be the cone of all positive semidefinite (symmetric) n × n real matrices. Matrices from Hn+(ℝ) play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on Hn+(ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on Hn+(ℝ). We characterize surjective, additive maps on Hn+(ℝ), n ≥ 3, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real n × n matrices that preserve the Löwner partial order in both directions.

Keywords: preserver; symmetric matrix; star partial order; minus partial order; Löwner partial order
MSC 2010: Primary 47B49; Secondary 15B48; 47L07; 54F05

The second author acknowledges the financial support from the Slovenian Research Agency, ARRS (research core funding No. P1-0288 and grants BI-TR/17-19-004 and BI-US/18-20-066).

  1. Communicated by Marcus Waurick

Acknowledgement

The authors wish to thank the anonymous referees for helpful and constructive comments that improved the presentation of this paper.

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Received: 2019-05-19
Accepted: 2019-12-02
Published Online: 2020-05-23
Published in Print: 2020-06-25

© 2020 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0386
Keywords for this article
preserver; symmetric matrix; star partial order; minus partial order; Löwner partial order

Articles in the same Issue

  1. Regular papers
  2. Recurrences for the genus polynomials of linear sequences of graphs
  3. The existence of states on EQ-algebras
  4. On sidon sequences of farey sequences, square roots and reciprocals
  5. Repdigits as sums of three balancing numbers
  6. Lipschitz one sets modulo sets of measure zero
  7. Refinement of fejér inequality for convex and co-ordinated convex functions
  8. An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
  9. Coefficients problems for families of holomorphic functions related to hyperbola
  10. Triebel-Lizorkin capacity and hausdorff measure in metric spaces
  11. The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
  12. On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
  13. Density of summable subsequences of a sequence and its applications
  14. Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
  15. A note on cosine series with coefficients of generalized bounded variation
  16. Some geometric properties of the non-Newtonian sequence spaces lp(N)
  17. On sequence spaces defined by the domain of a regular tribonacci matrix
  18. Jointly separating maps between vector-valued function spaces
  19. Some fixed point theorems for multi-valued mappings in graphical metric spaces
  20. Monotone transformations on the cone of all positive semidefinite real matrices
  21. Locally defined operators in the space of Ck,ω-functions
  22. Some properties of D-weak operator topology
  23. Estimating the distribution of a stochastic sum of IID random variables
  24. An internal characterization of complete regularity
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