向量在矩阵下的最小零化多项式

The Minimal Annihilation Polynomial of the Vector under a Matrix

给出了Ｃｎ中向量α在矩阵Ａ下的最小零化多项式ｄＡ，α（ｘ）的定义，并记ＬＡ（α）为由α，Ａα，Ａ２α，…生成的Ｃｎ的子空间，得到了如下结果：１．ｄＡ，α（ｘ）存在且唯一；２．ｄＡ，α（ｘ）的根都是Ａ的特征值；３．当α≠０时，ｄＡ，α（ｘ）无重根α可以表示成的Ａ不同特征值的特征向量之和；４．设ε１，ε２，…εｎ是Ｃｎ的一个基，则Ａ可以对角化ｄＡ，εｉ（ｘ）无重根，ｉ＝１，２，…，ｎ．

The paper has defined the concept of the minimal annihilation polynomial d A,α (x) of a vector α under a matrix A, and has denoted the generating subspace by the vectors α,Aα,A 2α,…,A n-1 α by L A(α).At last,the following results have been obtained:1.d A,α (x) is unique existence. 2.Every root of d A,α (x) is an eigenvalue of the matrix A. 3.Let α≠0.Then d A,α (x) has no multiple roots if and only if the sum of eigenvectors corresponds to the different eigenvalues of the matrix A. 4.Let ε 1,ε 2,…,ε n base of the lineor space C n and A∈M n(C).Then the matrix A is a diagonalization matrix if and only if no d A,ε i (x) has repeated roots for i=1,2,…,n.