数学物理学报(英文版) ›› 2005, Vol. 25 ›› Issue (3): 492-504.
储玉明
CHU Yu-Meng
摘要:
The main aim of this paper is to discuss the following two problems: \\
Problem I: Given $X\in H^{n\times m}$ (the set of all $n\times m$ quaternion matrices),
$\Lambda={\rm diag}(\lambda_1, \cdots$, $\lambda_m)\in H^{m\times m}$,
find $A\in BSH_\geq^{n\times n}$ such that $AX=X\Lambda$, where $BSH_\geq^{n\times n}$
denotes the set of all $n\times n$ quaternion matrices which are bi-self-conjugate and nonnegative definite.\\
ProblemⅡ: Given $B\in H^{n\times m}$, find $\overline{B}\in S_E$ such that
$
\|B-\overline{B}\|_Q=\min_{A\in S_E}\|B-A\|_{Q},
$
where $S_{E}$ is the solution set of problem Ⅰ, $\|\cdot\|_Q$ is the
quaternion matrix norm. The necessary and sufficient conditions for $S_E$ being nonempty are obtained.
The general form of elements in $S_E$ and the expression of the unique solution $\overline{B}$
of problem Ⅱ are given.
中图分类号: