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

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.