On a class of inverse palindromic eigenvalue problem

In this paper we first consider the following inverse palindromic eigenvalue problem (IPEP): Given matrices $ \Lambda = \text{diag}\{\lambda_{1}, \cdots, \lambda_{p}\} \in {\mathbb{C}}^{p\times p} $, $ \lambda_{i}\neq \lambda_{j} $ for $ i \neq j $, $ i, j = 1, \cdots, p $, $ X = [x_{1}, \cdots, x_{p}] \in {\mathbb{C}}^{n \times p} $ with $ \text{rank}(X) = p $, and both $ \Lambda $ and $ X $ are closed under complex conjugation in the sense that $ \lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}} $, $ x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n} $ for $ i = 1, \cdots, m $, and $ \lambda_{j} \in {\mathbb{R}} $, $ x_{j} \in {\mathbb{R}}^{n} $ for $ j = 2m+1, \cdots, p $, find a matrix $ A \in {\mathbb{R}}^{n \times n} $ such that $ AX = A^\top X\Lambda. $ We then consider a best approximation problem (BAP): Given $ \tilde{A} \in {\mathbb{R}}^{n \times n} $, find $ \hat{A} \in {\mathcal{S}}_{A} $ such that $ \|\hat{A}-\tilde{A}\| = \min_{{A} \in {\mathcal{S}_{A}}} \|A- \tilde{A}\|, $ where $ \|\cdot\| $ is the Frobenius norm and $ {\mathcal{S}}_{A} $ is the solution set of IPEP. By partitioning the matrix $ \Lambda $ and using the QR-decomposition, the expression of the general solution of Problem IPEP is derived. Also, we show that the best approximation solution $ \hat{A} $ is unique and derive an explicit formula for it.