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A matrix $P$ is said to be a nontrivial generalized reflection matrix overthe 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 generalizedantireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or$A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matricesof demension $n$. Let ${\large \varphi}$ be one of the following subsets of$\mathbb{H}^{n\times n}$ : (i) generalized reflexive matrix; (ii)reflexivematrix; (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 isto 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 $A.$\newlineIn 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 givenmatrix over $\mathbb{H}$\ and $\parallel \cdot\parallel_{F}$ is the Frobeniusnorm.

Inverse eigenproblems and approximation problems for the generalized  reflexive and antireflexive matrices with respect to a pair of generalized  reflection matrices