g-循环矩阵的对角化条件  

The diagonalization condition for g-circulant matrix

作　　者：周金土[1] 

机构地区：[1]浙江师范大学数理学院,浙江金华321004

出　　处：《浙江师范大学学报（自然科学版）》2004年第4期325-328,共4页Journal of Zhejiang Normal University:Natural Sciences

摘　　要：A是首行元素为a0,a1,…,an-1的n阶g 循环矩阵,f(x)=∑n-1i=0aixi.给出了用函数值f(εi)是否为零判别A可对角化的方法,这一方法有别于通常用线性无关特征向量个数的判别法,其中ε是一个n次原根.A matrix A, with first row a_0, a_1 ,…, a_(n-1) is said to be g-circulant, if it′s (i+1,(ig+j+1)) element is a_j, where i,j=0,1,…,n-1 and (ig+j+1)≡ig + j+1(mod n). ;It was proved the following theorem: ;Let A be a g-circulant matrix with first row, a_0,a_1,…,a_(n-1) and f(x)=∑n-1i=0a_ix^i. Let n_0=n,d_i=(g,n_(i-1)),n_(i-1)=d_in_i,i=1,2,…,k, where (g,n_k)=1. Let c=d_1d_2…d_(k-1),d=d_1d_2…d_k,m=n_k,n=dm,then A can be diagonalized if and only if all f(ε~i)=0, where c can not devides i, and when (f(ε~j)=0), with d|j, then all f(ε^(jg^t+sn_1))=0, where t=0,1,…,k-1, s=0,1,…,d_1-1, and k is the order of C={ig,ig^2,…ig^(k-1)} and ε is an n-th primitive root.

关 键 词：循环矩阵 可对角化 原根 线性无关 判别法 特征向量 函数值 个数 条件 方法 

分 类 号：O151.21[理学—数学] O156[理学—基础数学]