Vol.3 No.02(2014), Article ID:13503,6 pages DOI:10.12677/AAM.2014.32012

Yue Shen, Zhihui Wang, Kun Yan, Deyu Wu*

School of Mathematical Sciences, Inner Mongolia University, Hohhot

Email: *wudeyu2585@163.com

Received: Feb. 27th, 2014; revised: Mar. 29th, 2014; accepted: Apr. 9th, 2014

ABSTRACT

In this paper, we focus on the conditions under which the eigenvalues of complex Hamiltonian matrices are symmetric with respect to the real and imaginary axis, and the sufficient conditions that the eigenvalues of complex Hamiltonian matrices are the real or the pure imaginary number are obtained. In the end, a class of complex Hamiltonian matrices whose eigenvalues are symmetric with respect to the real and the imaginary axis are obtained.

Keywords:Eigenvalue, Eigenvector, Hamilton Matrix

Email: *wudeyu2585@163.com

1. 引言

2. 预备知识

1)的特征值关于虚轴对称；

2) 若。则有，即的特征值关于实轴和虚轴对称。

，代入上式得，然后等式两边右乘，结合，得到

2) 当时,，由于

.

，故矩阵相似，进而。再考虑到

.

3. 主要结果

1) 矩阵的特征值是实数或纯虚数，即

2)当且仅当，即的特征值关于原点对称。

.

1) 矩阵的特征值是实数或纯虚数，即

2)当且仅当，即的特征值关于原点对称。

，U是对应的特征向量，即有，则有

。证明完毕。

，我们同样可以得到

，如果，且对于任意的向量 (或，且)，那么

.

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NOTES

*通讯作者。