逻辑矩阵的特征向量及应用  

Eigenvector of Logical Matrix and Its Application

作　　者：李志强 秦军 宋金利 LI Zhi-qiang;QIN Jun;SONG Jin-li(School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China;Center of Henan Provincial Education Data Statistics and Analysis, Zhengzhou 450046, China)

机构地区：[1]河南财经政法大学数学与信息科学学院,河南郑州450046 [2]河南省教育统计数据分析和研究中心,河南郑州450046

出　　处：《数学的实践与认识》2018年第9期231-240,共10页Mathematics in Practice and Theory

基　　金：国家自然科学基金重点项目(61640315,61603125);河南省高等学校重点科研项目(18A110003);河南财经政法大学学术创新骨干支持计划;河南财经政法大学青年拔尖人才资助计划(hncjzfdxqnbjrc201607)

摘　　要：特征值与特征向量描述了线性变换的基本性质．特征向量是线性变换的作用下保持方向不变的向量，特征值体现了特征向量在线性变换中的伸缩性．讨论了一类布尔矩阵在布尔空间中的特征值与特征向量问题，证明了逻辑矩阵只有1特征值，所有1特征值构成1特征子空间，并且1特征子空间由唯一的一组基本特征向量布尔生成．最后，将逻辑矩阵特征向量的相关结果用于研究布尔网络极限环个数等拓扑性质．A linear transformation can be completely described by its eigenvalues and eigen- vectors. The eigenvalue denotes the flexibility of the eigenvectors under the transformation. The eigenvectors are vectors which are fixed in direction under a given linear transformation. The eigenvalues and eigenvectors of a special class of matrices, which is called logical（Boolean） matrix, are discussed in this paper. It is proved that the logical matrix have only 1-eigenvalue and eigenvector（s）, and the set of all the 1-eigenvectors is a subspace under Boolean plus and Boolean product. The subspace can be generated by one fundamental eigenvectors set under Boolean plus and Boolean product. At last, the results are used to discuss the topological property of Boolean network.

关 键 词：布尔矩阵 逻辑矩阵 特征向量 布尔网络 极限环 

分 类 号：O151.21[理学—数学]