IDEAS home Printed from https://ideas.repec.org/h/spr/spochp/978-3-030-27407-8_9.html
   My bibliography  Save this book chapter

Cauchy–Schwarz Inequality and Riccati Equation for Positive Semidefinite Matrices

In: Differential and Integral Inequalities

Author

Listed:
  • Masatoshi Fujii

    (Osaka Kyoiku University)

Abstract

By the use of the matrix geometric mean #, the matrix Cauchy–Schwarz inequality is given as Y ∗X ≤ X ∗X # U ∗Y ∗Y U for k × n matrices X and Y , where Y ∗X = U|Y ∗X| is a polar decomposition of Y ∗X with unitary U. In this note, we generalize Riccati equation as follows: X ∗A †X = B for positive semidefinite matrices, where A † is the Moore–Penrose generalized inverse of A. We consider when the matrix geometric mean A # B is a positive semidefinite solution of XA †X = B. For this, we discuss the case where the equality holds in the matrix Cauchy–Schwarz inequality.

Suggested Citation

  • Masatoshi Fujii, 2019. "Cauchy–Schwarz Inequality and Riccati Equation for Positive Semidefinite Matrices," Springer Optimization and Its Applications, in: Dorin Andrica & Themistocles M. Rassias (ed.), Differential and Integral Inequalities, pages 341-350, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-27407-8_9
    DOI: 10.1007/978-3-030-27407-8_9
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    More about this item

    Keywords

    47A64; 47A63; 15A09;
    All these keywords.

    JEL classification:

    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:spr:spochp:978-3-030-27407-8_9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.