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. ***
