The Leslie Matrix Solution of the Reduced Biquaternion Matrix Equation

AXB+CXD=E

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB+CXD=E$$\end{document}

Abstract

This paper investigates two different Leslie matrix solutions for the reduced biquaternion matrix equation

AXB+CXD=E

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB+CXD=E$$\end{document}

. Through the permutation matrices, the complex decomposition of reduced biquaternion matrices, and the Kronecker product, by leveraging the specific attributes of Leslie matrices, we transform the constrained reduced biquaternion matrix equation into an unconstrained form. Consequently, we derive the necessary and sufficient conditions for the existence of solutions in the form of Leslie matrices to the reduced biquaternion matrix equation

AXB+CXD=E

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB+CXD=E$$\end{document}

and provide a general expression for such solutions. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.

Keywords

Reduced biquaternion / Leslie matrix / Kronecker product / Complex decomposition / Optimal approximation / 15A09 / 15B33

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Funding

the Sichuan Science and Technology Program(No. 2024NSFSC0721)

the 2024 School-Level Project of Guangxi Financial Vocational College(No. 607001)

the National Natural Science Foundation of China(No. 12361078)

the Postdoctoral Fellowship Program of CPSF(No. GZB20230092)

the China Postdoctoral Science Foundation(No. 2023M740383)

RIGHTS & PERMISSIONS

Shanghai University