The generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation

Abstract

Let $ J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n} $. A matrix $ A \in \mathbb{R}^{2n\times 2n} $ is said to be Hamiltonian if $ (AJ)^{\top} = AJ $. In this paper, we first consider the following generalized inverse eigenvalue problem (GIEP): Given a pair of matrices $ (\Lambda, X) $ in the form $ \Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p} $ and $ X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p} $, where diagonal elements of $ \Lambda $ are all distinct with 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}, $ $ {\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n} $ for $ i = 1, \cdots, l, $ and $ \lambda_{j}\in \mathbb{R}, $ $ {\bf{x}}_{j}\in \mathbb{R}^{2n} $ for $ j = 2l+1, \cdots, p. $ Find Hamiltonian matrices $ A $ and $ B $ such that $ AX\Lambda = BX. $ Then, we consider the associated optimal approximation problem (OAP): Given $ \tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n} $. Find $ (\hat{A}, \hat{B}) \in \mathbb{S_{E}} $ such that $ \|\hat{A}-\tilde{A}\|^2+\|\hat{B}-\tilde{B}\|^2 = {\min_{(A, B)\in \mathbb{S_{E}}}}\left(\|A-\tilde{A}\|^2+\|B-\tilde{B}\|^2\right), $ where $ \mathbb{S_{E}} $ is the solution set of Problem GIEP. By using the QR-decomposition, we deduce the representation of the general solution of Problem GIEP. Also, we obtain the unique optimal approximation solution $ (\hat{A}, \hat{B}) $ of Problem OAP.

Keywords

• generalized inverse eigenvalue problem
• hamiltonian matrix
• qr-decomposition
• optimal approximation