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

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A matrix P is said to be a nontrivial generalized reflection matrix over the real quaternion algebra H if P=P≠ I and P²=I where means conjugate and transpose. We say that Aⁿ× 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 φ be one of the following subsets of Hⁿ× ⁿ: (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let Zⁿ× ᵐ with rank(Z) =m and Λ= diag(λ₁,...,λₘ). The inverse eigenproblem is to find amatrix A such that the set φ(Z,Λ) ={ A∈ φ | AZ=ZΛ} nonempty and find the general expression of A. In this paper, we investigate the inverse eigenproblem φ(Z,Λ). Moreover, the approximation problem: A∈ φ A-E F is studied, where E is a given matrix over Hand ∥ ·∥F is the Frobenius norm.