Research On Certain Tridiagonal Matrix

Tridiagonal matrices are an important kind of special matrices with a wide range of applications in engineering, medicine and signal processing. In particular, solving difference equation, differential equation and delay differential equation, we often need to compute powers and inverse of tridiagonal matrices. Recently, people are interested in studying some properties of tridiagonal matrices because of their wide application. In this paper, we mainly study related problems of two kinds of complex tridiagonal matrices, such as determinant, eigenvalue problem, explicit powers and so on. The main content of this thesis is arranged as follows:In the first chapter, we first introduce research background of tridiagonal matrices, the present research situation of the topic at home and abroad. Then we give the definitions and properties of Chebyshev polynomials and some results in matrix theory, which are preliminaries for the study of the properties of tridiagonal matrices. Finally, we state the main work of this paper, innovative points and difficult points.In the second chapter, a new method for computing determinant, characteristic polyno-mial, eigenvalues and eigenvectors of complex tridiagonal matrices is proposed. This method is not only different from the method of symbolic calculus of semi-infinite sequences used by Yueh and Willms but also differs from the technique based on theory of recurrence sequences adopted by Kouachi. The technique of this method is connecting Chebyshev polynomial-s with the determinant of tridiagonal matrices and computing determinant, characteristic polynomial, eigenvalues and eigenvectors of complex tridiagonal matrices on the basis of properties of Chebyshev polynomials. We consider two kinds of tridiagonal matrices in this chapter. The first one is a kind of complex tridiagonal matrices with the first and last superdiagonal entries and subdiagonal entries being variable. The second one is a kind of complex tridiagonal matrices with the first and last main diagonal entries being variable and the constant products of subdiagonal entries and the corresponding superdiagonal entries. Through calculating determinant, characteristic polynomial, eigenvalues and eigenvectors of these two kinds of tridiagonal matrices, we present this new method completely.In the third chapter, we mainly consider the second kind of tridiagonal matrices. Firstly, we present and prove the spectral decomposition of the second kind of tridiagonal matrices through constructing the inverse matrix of the transformation matrix. Secondly, we compute the arbitrary integer powers and explicit inverse of the second kind of tridiagonal matrices according to the spectral decomposition. Finally, we give the condition under which the sec-ond kind of tridiagonal matrices can be unitarily diagonalizable and prove some conclusions about families of the second kind of tridiagonal matrices.In the fourth chapter, we summarize the main ideas and content of this paper and give some constructive opinions.