The Toeplitz Matrices Applied In Time-Series Analysis

This thesis focuses on one important and special structured matrix—Toeplitz matrices and its application in Time-series analysis .Study on Toeplitz matrices are not only an important part of matrix and computational theory, but also an active and significant research direction in the field of Application Math. Linear equations Tx = b is Toeplitz system when T is Toeplitz matrix, it is used in various fields, including the information and graph management , numerical solution of differential equation and integral equation , queuing theory and cybernetics theory, statistics theory, ect. The systematic study of its valid calculating methods is of great significance and application value. Since this kind of matrix has many excellent qualities and structure, it is necessary to probe into its special qualities, special structure, eigenvalue problem and inverse eigenvalue problem. Toeplitz matrix is a peripheral subject between the fields of Linear Algebra and Functional Analysis. Thesis restricts its content in the area of Toeplitz, focuses on asymptotic behavior of eigenvalue and proposes its application in Tims-Series Analysis.The arrangement of the thesis is shown as following:Chapter 1 mainly introduces the application of Toepllitz matrix in Tims-Series Analysis.Chapter 2 introduces some qualities of matrix: first gives the definition of asymptotic behavior of matrix, then the qualities of it's: theorem 2.1 2.2 2.4 2.5.Chapter 3 introduces relative qualities of the circulant matrices.Chapter 4 discusses the qualities of Asymptotically Equivalent Toeplitz matrices, Asymptotically Equivalent relative between Toeplitz matrices and circulant matrices, relative between eigencalue Toeplitz matrices and Fourier series.