向量在矩阵下的最小零化多项式与复矩阵A的特征向量的关系

The Relations between the Ninimal Annihilation Polynomial of Vector in Matric And the Characteristic of Complex Matrix A

给出了 Cn中向量α在矩阵 A下的最小零化多项式 d A,α( o)的定义 ,并记 LA( α)为由 α,Aα,A2 α,…生成 Cn的子空间 ,得到了如下结果 :1 .存在Φ∈LA( α) ,Φ≠ 0和数 λ使得 AΦ=λΦ d A,α( λ) =0 ;2 .LA( α)中属于 A的特征值λ的特征向量一般表示式 ;3.当α≠ 0时 ,d A,α( x)无重根 α可以表示成 A的不同特征值的特征向量之和 ;4.存在α∈ Cn,使得 A的每一特征向量都属于 LA( α) A的特征子空间都是一维的 ;

In this paper,we have defined the concept that the minimal annihilation polynomial d A,α (X) of a vector a under a matrix A and we denote the generating subspace spanned by the vectors α,Aα,A 2α,… by L A(α).At last the following results have been obtained: 1.There exists a vector Φ∈L A(α),Φ≠0 and λ∈C n such that A Φ=λΦ if and only of d A,α (λ)=0 2.We have obtained the general form of the eigenvector corresponding to the eiqenvalue λ in L A(α). 3.Let α≠0.Then d A,α (X) has no multiple roots if and only if the vector a has a decomposition into the sum of eigenvalues corresponding to the different eigenvalues of the matrisx A. 4.There exists a vector α∈C n,such that all the eigenvectors of the matrix A are in L A(α) if and only if the dimension of every eiqensubspace of the matrix A is one.