Mathematical Sciences: RUI: Spectral Properties of Structured Matrices

9305856沟槽如果将一个方阵的行和列的顺序颠倒会产生矩阵的转置，那么这个方阵就是超对称的。Toeplitz矩阵是最重要的超对称矩阵，其中沿平行于主对角线的每个条纹上的元素是常数。反转厄米图普利兹矩阵的特征向量的元素顺序产生特征向量的共轭。一个2n （2n+1）阶的实对称矩阵有n （n+1）个线性无关的特征向量，使得反转其元素的顺序使特征向量保持不变，并且有n个线性无关的特征向量，使得反转元素的顺序相当于将特征向量乘以-1。与这两种特征向量相关联的特征值集合分别称为矩阵的偶谱和奇谱。以特征向量的元素为系数的多项式称为矩阵的特征多项式。研究了实对称Toeplitz矩阵的偶谱和奇谱的互交性质、厄米特Toeplitz矩阵的特征多项式的零点位置、实对称Toeplitz矩阵的特征值反问题的数值解，以及更一般的厄米特过对称矩阵和实对称过对称矩阵的类似问题。这里考虑的结构化矩阵在统计学和信号处理中有重要的应用。在这些应用中出现的矩阵通常非常大。本研究试图利用这些矩阵的特殊结构，以有效地解决这些领域的问题。研究结果的主要应用将是在信号处理领域，即从传输信号中提取信息的科学。＊＊＊

9305856 Trench A square matrix is said to be persymmetric if reversing the orders of its rows and columns produces the transpose of the matrix. Toeplitz matrices, in which the elements along each stripe parallel to the main diagonal are constant, are the most important persymmetric matrices. Reversing the order of the elements of an eigenvector of a Hermitian Toeplitz matrix produces the conjugate of the eigenvector. A real symmetric matrix of order 2n (2n+1) has n (n+1) linearly independent eigenvectors such that reversing the order of their elements leaves the eigenvectors unchanged, and n linearly independent eigenvectors such that reversing the order of the elements is equivalent to multiplying the eigenvector by -1. The sets of eigenvalues associated with these two kinds of eigenvectors are called the even and odd spectra of the matrix, respectively. The polynomials having the elements of eigenvectors as coefficients are called the eigenpolynomials of the matrix. The investigator studies the interlacement properties of the even and odd spectra of real symmetric Toeplitz matrices, the location of the zeros of the eigenpolynomials of Hermitian Toeplitz matrices, numerical solution of the inverse eigenvalue problem for real symmetric Toeplitz matrices, and analogous problems for more general Hermitian persymmetric matrices and real symmetric persymmetric matrices. Structured matrices of the kind considered here have important applications in statistics and signal processing. The matrices occurring in these applications are usually very large. This research attempts to exploit the peculiar structure of these matrices in order to solve problems in these areas efficiently. The main applications of the results will be in the area of signal processing, the science of extracting information from transmitted signals. ***