箭状矩阵的广义特征值反问题  

Generalized Inverse Eigenvalue Problem for Arrow-Like Matrices

作　　者：殷庆祥[1] 

机构地区：[1]盐城师范学院数学系,盐城224002

出　　处：《南京航空航天大学学报》2002年第2期190-192,共3页Journal of Nanjing University of Aeronautics & Astronautics

摘　　要：讨论实对称箭状矩阵 (除对角元及最后一行、最后一列元素外 ,其余位置元素全为零 )的广义特征值反问题 ,它可以用来描述星形弹簧质量系统的振动问题 ,即给出系统的振动频率如何来确定质点的质量或弹簧的刚度。通过对箭状矩阵特征多项式性质的研究 ,运用部分分式理论 ,证明了给定正定箭状矩阵 B,实数 {λi}ni=1 ,{μi}n- 1 i=1 ,满足λ1 <μ1 <… <μn- 1 <λn,存在箭状矩阵 A,使广义特征值问题 Ax=λBx有解 {λi}ni=1 ,而广义特征值问题 A(n-1 ) x=λB(n-1 ) x有解 {μi}n- 1 i=1 ,其中 A(n-1 ) ,B(n-1 )分别表示 A,B的 n-1级主子矩阵。The generalized inverse eigenvalue problem for real symmetric arrow like matrices (the matrices whose elements are all zero except for those laying in the diagonal and the last row and column) is dealt with which can describe the vibration problem of star like spring mass system, i.e.,how to determine the mass of particle or the stiffness of spring. With a thorough study of the characteristic polynomial of a arrow like matrix, using the theory of partial fraction, it may prove: given a positive arrow like matrix B, real numbers { λ i} n i=1 ,{μ i} n-1 i=1 satisfying λ 1<μ 1<...<μ n-1 <λ n ,there exists a arrow like matrix A so that the generalized eigenvalue problem Ax=λBx has solution { λ i} n i=1 and the generalized eigenvalue problem A(n-1)x=λB(n-1)x has solution { μ i} n-1 i=1 ,where A(n-1),B(n-1) denote the n-1 order leading principal matrix of A,B, respectively.

关 键 词：箭状矩阵 广义特征值 反问题 实对称矩阵 

分 类 号：O241.6[理学—计算数学]