Inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their optimal approximations
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Inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their optimal approximations Wei-Ru Xu1
· Guo-Liang Chen2,3
Received: 10 March 2020 / Revised: 30 April 2020 / Accepted: 22 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, the inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their associated optimal approximation problems which are constrained by their partially prescribed eigenpairs are considered, respectively. First, the necessary and sufficient conditions of the solvability for the inverse eigenvalue problems of skew-Hermitian reflexive and anti-reflexive matrices are both derived, and the general solutions are also presented. Then the solutions of the corresponding optimal approximation problems in the Frobenius norm to a given matrix are also given, respectively. Furthermore, we give the algorithms to compute the optimal approximate skew-Hermitian reflexive and anti-reflexive solutions and present some illustrative numerical examples. Keywords Inverse eigenvalue problem · Optimal approximation problem · Skew-Hermitian reflexive matrix · Skew-Hermitian anti-reflexive matrix Mathematics subject classification 65F18 · 15A51 · 15A18 · 15A12
1 Introduction Throughout this paper, let C n×m , HC n×n and SHC n×n be the sets of all complex matrices, Hermitian matrices and skew-Hermitian matrices, respectively. Denote by AH , A† the conjugate transpose and Moore-Penrose generalized inverse of a matrix A ∈ C n×m , respectively.
Communicated by Jinyun Yuan.
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Wei-Ru Xu [email protected],[email protected] Guo-Liang Chen [email protected]
1
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, People’s Republic of China
2
School of Mathematical Sciences, East China Normal University, Shanghai 200241, People’s Republic of China
3
Shanghai Key Laboratory of PMMP, Shanghai 200241, People’s Republic of China 0123456789().: V,-vol
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W.-R. Xu, G.L. Chen
In signifies the identity matrix of order n. P ∈ C n×n is referred to as a generalized reflection matrix if P satisfies P H = P and P 2 = In , i.e., P is an involutory Hermitian matrix. In the following, we give the definitions of reflexive and anti-reflexive matrices. Definition 1.1 (Chen 1998) Given a generalized reflection matrix P ∈ C n×n . 1. A matrix A ∈ C n×n is referred to as a reflexive matrix if P A P = A. The set of all n × n reflexive matrices is denoted by Crn×n (P). 2. A matrix A ∈ C n×n is referred to as an anti-reflexive matrix if P A P = −A. The set of all n × n anti-reflexive matrices is denoted by Can×n (P). It is clear to find that all matrices in the sets Crn×n (P) and Can×n (P) depend on the matrix P. If P is the cross-identity matrix of order n, then the sets Crn×n (P) and Can×n (P) in real number field are the well-known sets of centrosymmetric and skew-centrosymmetric matrices, respectively. Definition 1.2 Given a generalized reflection matrix P ∈ C
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