Abstract
Let J in{mathbb{R}}^{ntimes n} be a normal matrix such that J^{2}=-I_{n}, where I_{n} is an n-by-n identity matrix. In (S. Gigola, L. Lebtahi, N. Thome in Appl. Math. Lett. 48:36–40, 2015) it was introduced that a matrix A in{mathbb {C}}^{ntimes n} is referred to as normal J-Hamiltonian if and only if {(AJ)}^{*}=AJ and AA^{*}=A^{*}A. Furthermore, the necessary and sufficient conditions for the inverse eigenvalue problem of such matrices to be solvable were given. We present some alternative conditions to those given in the aforementioned paper for normal skew J-Hamiltonian matrices. By using Moore–Penrose generalized inverse and generalized singular value decomposition, the necessary and sufficient conditions of its solvability are obtained and a solvable general representation is presented.
Highlights
1 Introduction In this paper, we mainly discuss the following partially described inverse eigenvalue problem which is considered in linear manifold
The above problem usually appears in the design and modification of mass-spring systems, dynamic structures, Hopfield neural networks, vibration in mechanic, civil engineering, and aviation [2,3,4]
Bai [6] settled the case of Hermitian and generalized skew-Hamiltonian matrices
Summary
We mainly discuss the following partially described inverse eigenvalue problem which is considered in linear manifold. Zhang et al [5] solved the inverse eigenvalue problem of Hermitian and generalized Hamiltonian matrices. Definition 2 Let J ∈ Rn×n be a normal matrix such that J2 = –In. A matrix A ∈ Cn×n is called normal skew J-Hamiltonian if and only if (AJ)∗ = –AJ and AA∗ = A∗A. We present a set of alternative conditions assuring the solvability of the problem that involves skew normal J-Hamiltonian matrices. In order to present more simple conditions to be verified, we mainly use Sun’s [10] and Penrose’s [11] results and the generalized singular value decomposition to solve Problem 1 when the set = N Sn×n(J). A similar technique may be used to solve the inverse eigenvalue problem for normal J-Hamiltonian matrices
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