利用矩阵的四分块形式刻画了M自共轭矩阵的特征结构,并讨论了四元数广义Sylvester方程AX-YB=C的一类M自共轭混合结构解,其中X为酉相似块对角M自共轭矩阵,Y为自共轭矩阵.根据所提结构矩阵的特点,将原方程转化为等价的无约束方程组,再利用矩阵的Moore-Penrose广义逆,获得方程组可解的充分必要条件及其通解表达式,从而得到原方程的M自共轭混合结构解.特别地,导出矩阵方程AX=C具有酉相似块对角M自共轭解的充要条件及其通解表达式.当M=0时,利用四元数矩阵对的CCD-Q分解,获得广义Sylvester方程满足‖Y~‖=min‖Y‖的约束混合结构解集.数值算例检验了所得结果的正确及可行性.
In this paper, using the specific structure M self-conjugate matrix in the form of quarto-block matrix, we discuss a class of M self-conjugate mixed structure solutions of generalized Sylvester equation AX-YB=C on the quaternion field, where X is a unitary similar block diagonal M self-conjugate matrix, Y is a self-conjugate matrix. According to the characteristics of the proposed structure matrix, the original structural equation is transformed into an equivalent system of unconstrained set of equations, and the necessary and sufficient conditions for the solution of the equations and the general solution expression are obtained by using the Moore-Penrose generalized inverse of the matrix. Thus, the M self-conjugate mixed structure solution of the original equation is obtained. In particular, the necessary and sufficient conditions for matrix equation AX=C to have unitary similar block diagonal M self-conjugate solutions and the general solution expression are derived. When M=0, the constrained mixed structure solution set of generalized Sylvester equation satisfying ‖Y~‖=min‖Y‖ is obtained by using the CCD-Q decomposition of quaternion matrix pairs. A numerical example is given to verify the correctness and feasibility of the results.- DOI:10.3969/j.issn.1000-1565.2023.04.001四元数广义Sylvester方程M自共轭混合结构解徐云,黄敬频(广西民族大学 数学与物理学院,广西 南宁 530006)摘 要:利用矩阵的四分块形式刻画了M自共轭矩阵的特征结构,并讨论了四元数广义Sylvester方程AX-YB=C的一类M自共轭混合结构解,其中X为酉相似块对角M自共轭矩阵,Y为自共轭矩阵.根据所提结构矩阵的特点,将原方程转化为等价的无约束方程组,再利用矩阵的Moore-Penrose广义逆,获得方程组可解的充分必要条件及其通解表达式,从而得到原方程的M自共轭混合结构解.特别地,导出矩阵方程AX=C具有酉相似块对角M自共轭解的充要条件及其通解表达式.当M=0时,利用四元数矩阵对的CCD-Q分解,获得广义Sylvester方程满足‖Y~‖=min‖Y‖的约束混合结构解集.数值算例检验了所得结果的正确及可行性.关键词:四元数;广义Sylvester方程;M自共轭;混合结构解中图分类号:O241.6 文献标志码:A 文章编号:1000-1565(2023)04-0337-09M self-conjugate mixed structure solutions of the quaternion generalized Sylvester equationXU Yun,HUANG Jingpin(College of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China)Abstract: In this paper, using the specific structure M self-conjugate matrix in the form of quarto-block matrix, we discuss a class of M self-conjugate mixed structure solutions of generalized Sylvester equation AX-YB=C on the quaternion field, where X is a unitary similar block diagonal M self-conjugate matrix, Y is a self-conjugate matrix. According to the characteristics of the proposed structure matrix, the original structural equation is transformed into an equivalent system of unconstrained set of equations, and the necessary and sufficient conditions for the solution of the equations and the general solution expression are obtained by using the Moore-Penrose generalized inverse of the matrix. Thus, the M self-conjugate mixed structure solution of the original equation is obtained. In particular, the necessary and sufficient conditions for matrix equation AX=C to have unitary similar block diagonal M self-conjugate solutions and the general solution expression are derived. When M=0, the constrained mixed structure solution set of generalized Sylvester equation satisfying ‖Y~‖=min‖Y‖ is obtained by using the CCD-Q decomposition of quaternion matrix pairs. A numerical example is given to verify the correctness and feasibility of the results.- 收稿日期:2022-10-07 基金项目:国家自然科学基金资助项目(11661011) 第一作者:徐云(1996—), 女, 贵州水城人, 广西民族大学在读硕士研究生, 主要从事数值代数方向研究.E-mail:960455313@qq.com 通信作者:黄敬频(1964—), 男, 广西陆川人, 广西民族大学教授, 主要从事数值代数方向研究. E-mail:hjp2990@126.com第4期徐云等:四元数广义Sylvester方程M自共轭混合结构解
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