Abstract:

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.
 

Key words: Conjugate, inverse eigenvalue problem, quaternion matrix