实对称五对角矩阵Procrustes问题

本文研究了实对称五对角矩阵Procrustes.利用矩阵的奇异值分解简化问题,得到了实对称五对角矩阵X极小化,最后给出数值算例说明方法的有效性.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

实对称五对角矩阵逆特征值问题

1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问     

一类特殊五对角和七对角行列式的计算

采用递推法研究了对称和非对称五对角行列式的通项公式.对于对称情形,给出显式表达式;对于非对称情形,通项公式为六个指数函数的线性组合.还得到对称七对角行列式的通项公式.     

五对角线矩阵的逆特征值问题

研究了由两个或三个有序的特征值和特征向量来构造规范五对角线矩阵的逆特征值问题,并给出了问题的解的表达式和一些具体的数值例子.     

三对角与五对角Toeplitz矩阵求逆的算法

提出了一种求三对角与五对角Toeplitz矩阵逆的快速算法,其思想为先将Toeplitz矩阵扩展为循环矩阵,再快速求循环矩阵的逆,进而运用恰当矩阵分块求原Toeplitz矩阵的逆的算法.算法稳定性较好且复杂度较低.数值例子显示了算法的有效性和稳定性,并指出了算法的适用范围.     

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r_k}_k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r_k}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.