四元数矩阵方程 AX+ X ∗ B+CY=D的三对角广义(反)对称解
Tridiagonal Generalized (Skew-) Symmetric Solution for the Quaternion Matrix Equation AX+ X ∗ B+CY=D
摘要: 文章对于给定的四元数矩阵 A,B,C 和 D ，深入讨论了矩阵方程 AX+ X ∗ B+CY=D 的三对角广义(反)对称解。利用Kronecker积，矩阵拉直算子以及Moore-Penrose广义逆等理论，充分考虑三对角广义(反)对称矩阵的结构特点，讨论了四元数矩阵方程三对角广义(反)对称解的结果，给出方程有解的充分必要条件及解的表达式。
Abstract: This paper discusses in depth the tridiagonal generalized (skew-) symmetric solutions for the given quaternion matrices A,B,C , and D in the matrix equation AX+ X ∗ B+CY=D . By employing the theories of the Kronecker product, matrix vectorization, and the Moore-Penrose generalized inverse, the research thoroughly considers the structural characteristics of tridiagonal generalized (skew-) symmetric matrices. It discusses the outcomes of the quaternion matrix equation’s tridiagonal generalized (skew-) symmetric solutions, provides the necessary and sufficient conditions for the equation to have a solution, and presents the expressions for these solutions.
文章引用：王梓沣, 张澜. 四元数矩阵方程 AX+ X ∗ B+CY=D的三对角广义(反)对称解[J]. 应用数学进展, 2025, 14(2): 251-262. https://doi.org/10.12677/aam.2025.142068

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