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

TL;DR

This paper investigates inverse eigenproblems and approximation problems for generalized reflexive and antireflexive matrices over quaternions, providing explicit solutions and characterizations for these matrix classes.

Contribution

It introduces new inverse eigenproblem solutions and approximation methods for generalized reflexive and antireflexive matrices over quaternion algebra.

Findings

Derived explicit formulas for matrices satisfying the inverse eigenproblem.

Established conditions for the nonemptiness of the solution set.

Developed approximation techniques minimizing Frobenius norm differences.

Abstract

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.