| Summary: | Abstract Let J∈Rn×n $J \in{\mathbb{R}}^{n\times n}$ be a normal matrix such that J2=−In $J^{2}=-I_{n}$, where In $I_{n}$ 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∈Cn×n $A \in{\mathbb {C}}^{n\times n}$ is referred to as normal J-Hamiltonian if and only if (AJ)∗=AJ ${(AJ)}^{*}=AJ$ and AA∗=A∗A $AA^{*}=A^{*}A$. 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.
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