The Inverse Eigenvalue Problems For Hermitian Toeplitz Matrix

Hermitian Toeplitz matrices make an important role in the the stochastic filtering,the signal processing, the biological information processing, trigonometric moment prob-lem, the Szego¨ theory and other engineering problems. Hermitian Toeplitz matrices arean important subclass of structured matrices. Any Hermitian Toeplitz matrix Hncan berewritten as the addition of a real symmetric Toeplitz matrix A and a real skew-symmetricToeplitz matrix B. That is Hn=A+iB, and i is the the imaginary unit which meansi2=1. And if we reorder the eigenvalue set of the two matrices, the addition of the twoeigenvalue set is just the eigenvalue set of matrix Hnwhat is given. In this paper, weapply the eigenvalue set what is given to reconstruct a Hermitian Toeplitz matrix from theknowledge of interpolation.This paper also considers to reconstruct a Hermitian Toeplitz matrix Hnfrom thegiven eigenpairs. When only one eigenpair or two eigenpairs are given, the existence ofthe solution of this problem is proposed. We find that only one Hermitian Toeplitz matrixexists from the two prescribed eigenpairs.