在本文中，我們討論Toeplitz矩陣的特徵值反問題。首先，我們將由簡單的代數方法引入此問題的困難度。其次，我們將介紹矩陣特徵值在幾何上所表示的意義，並設法由此來猜想解的存在與唯一性的問題。然而，對於高維度的問題，牛頓法依舊是最有效率的，然而其起始值的取法卻不甚容易，因此引入一取起始值的方法以供牛頓法使用。

In this thesis, we consider the inverse Toeplitz eigenvalue problem which recover a real symmetric Toeplitz with desired eigenvalues. First some lower dimensional cases are solved by algebraic methods. This gives more insight on the inverse problem. Next, we explore the geometric meaning of real symmetric Toeplitz matrices. For high dimensional cases, numerical are unavoidable. From our numerical experiments, Newton-like methods are very effective for this problem.

1. Introduction
2. Algebraic methods
3. Geometric Viewpoint
4. Newton-like method
5. Good initial value
6. Convergent Experiments
7. Concluding Remarks

1. Dirk Laurie, The initial values for the inverse toeplitz eigenvalue problem.
2. Raymond H. Chan, H. L. Chung and Shu-Fang Xu, The inexact newton-like method for inverse toeplitz eigenvalue problem.