IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i3p393-d1576616.html
   My bibliography  Save this article

Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures

Author

Listed:
  • Danyal Ahmad

    (NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad 44000, Pakistan)

  • Muhammad Faisal Khan

    (Department of Science and High Technology, University of Insubria, 22100 Como, Italy)

  • Stefano Serra-Capizzano

    (Department of Science and High Technology, University of Insubria, 22100 Como, Italy
    Department of Information Technology, University of Uppsala, 75310 Uppsala, Sweden)

Abstract

In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) ∗-algebra. We consider the geometric mean for two matrices, using the Ando-Li-Mathias (ALM) definition, and then we pass to the extension of the idea to more than two matrices by introducing the Karcher mean. While there is no simple formula for the geometric mean of more than two matrices, iterative methods from the literature are employed to compute it. The main novelty of the work is the extension of the study in the distributional sense when input matrix-sequences belong to one of the GLT ∗-algebras. More precisely, we show that the geometric mean of more than two positive definite GLT matrix-sequences forms a new GLT matrix-sequence, with the GLT symbol given by the geometric mean of the individual symbols. Numerical experiments are reported concerning scalar and block GLT matrix-sequences in both one-dimensional and two-dimensional cases. A section with conclusions and open problems ends the current work.

Suggested Citation

  • Danyal Ahmad & Muhammad Faisal Khan & Stefano Serra-Capizzano, 2025. "Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures," Mathematics, MDPI, vol. 13(3), pages 1-27, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:393-:d:1576616
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/3/393/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/3/393/
    Download Restriction: no
    ---><---

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:393-:d:1576616. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.