证明Cayley—Hamilton定理——兼谈矩阵解析函数的围道积分表示

Proof of Cayley—Hamilton Theorem On the Representation of a Contour Integral of the Matrix Analytic Function

本文用简易法证明两个矩阵定理Cayley—Hamilton定理矩阵A的特征多项式f(λ)=|λI-A|=λ~n+a_1λ^(n-1)+…+a_n则有f(A)=A^n+a_1A^(n-1)+…+a_nI=0定理解析函数的柯西公式拓宽成矩阵解析函数的围道积分表示如下:f(A)=1/(2πi)∮_c(f(z)/(zI)-A)(dz)

In this paper we prove two matrix theorems in a simple way.Cayley-Hamilton theorem Matrix A,characteristic polynomial:f(λ)=|λI-A|=λ^n+a1λ^n-1+…+an Then f(A)=A^n+a 1 A^n-1+…+anI=0 Theorem:The Cauchy formula of the analytic function is extended to the representation of a Contour integral of the matrix analytic function.f(A)=1/2πic∮f(z)zI-Adz.