Journal of East China Normal University(Natural Sc ›› 2018, Vol. 2018 ›› Issue (4): 32-46,58.doi: 10.3969/j.issn.1000-5641.2018.04.004

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The skew-Hermitian {P,k+1} Hamiltonian solutions of a linear matrix equation

YONG Jin-jun1,2, CHEN Guo-liang2, XU Wei-ru2   

  1. 1. Department of Mathematics and Computer Science, Guizhou Education University, Guiyang 550018, China;
    2. School of Mathematical Sciemces, East China Normal University, Shanghai 200241, China
  • Received:2017-07-19 Online:2018-07-25 Published:2018-07-19

Abstract: Given PCn×n and P*=-P=P k+1, we consider the necessary and sufficient conditions such that the matrix equation AX=B is consistent with the skew-Hermitian {P, k + 1} (skew-) Hamiltonian structural constraint. Then, the corresponding expressions of the constraint solutions are also obtained. For any given matrix à ∈ Cn×n, we present the optimal approximate solution ā ∈ Cn×n such that ||Ã-ā|| is minimized in the Frobenius norm sense. If the matrix equation AX=B is not consistent, its least-squares skew-Hermitian {P, k + 1} (skew-) Hamiltonian solutions are given. Under the least-square sense, we consider the best approximate solutions to any given matrix. Finally, some illustrative experiments are also presented.

Key words: skew-Hermitian matrix, Hamiltonian matrix, least-squares solution, skew-Hermitian{P,k+1}Hamiltonian matrix

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