A matrix $P$ is said to be a nontrivial generalized reflection matrix over the real quaternion algebra $\mathbb{ H}$ if $P^{ \ast }=P\neq I$ and $P^{ 2}=I$ where $\ast$ means conjugate and transpose. We say that $A\in\mathbb{ H}^{ n\times n}$ is generalized reflexive (or generalized antireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or $A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matrices of demension $n$. Let ${ \large \varphi}$ be one of the following subsets of $\mathbb{ H}^{ n\times n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let $Z\in\mathbb{ H}^{ n\times m}$ with rank$\left( Z\right) =m$ and $\Lambda=$ diag$\left( \lambda_{ 1},...,\lambda_{ m}\right) .$ The inverse eigenproblem is to find a\ matrix $A$ such that the set ${ \large \varphi }\left( Z,\Lambda\right) =\left\{ A\in{ \large \varphi}\text{ }|\text{ }AZ=Z\Lambda\right\} $ nonempty and find the general expression of $A.$\newline In this paper, we investigate the inverse eigenproblem ${ \large \varphi}\left( Z,\Lambda\right) $. Moreover, the approximation problem: $\underset{ A\in{ \large \varphi}}{ \min\left\Vert A-E\right\Vert _{ F}}$ is studied, where $E$ is a given matrix over $\mathbb{ H}$\ and $\parallel \cdot\parallel_{ F}$ is the Frobenius norm.
Categories: math.RA
