防御矩阵满足乘法交换律的证明

Study on the Defense Matrix Meeting Commutative Law of Multiplication

在使用马尔柯夫链分析多层防御系统的防御效用值时,发现防御矩阵是否满足乘法交换律将关系到多层防御系统变换部署后的防御效用值,因此有必要对防御矩阵是否满足乘法交换律进行证明。首先介绍了防御矩阵的概念、物理意义、重要性质及计算方法,分析了防御矩阵满足乘法交换律的重要意义,最后综合运用数学归纳法和随机矩阵性质证明了防御矩阵满足乘法交换律的事实,此结论无论对于多层防御系统的防御效用值研究还是矩阵理论研究都有一定的指导作用。

The concept and physical meaning of defense matrix is introduced firstly. The method of computing defense matrix is also presented. The significance of the fact that defense matrix meet commutative law of multiplication is analyzed. Finally, the fact is proved using mathematical induction and the character of stochastic matrix. The result can offer direction to not only research on defense effectiveness of multilayer defense system but also research on the theory of matrix.