On The Numerical Ranges Of Some Tridiagonal Matrices

Tridiagonal matrices as a kind of special matrices are widely used in various fields,especially in solving difference equations and a system of linear equations. It is required to compute the inverse and power of tridiagonal matrices. The uses of spline interpolation and other particular methods need to rely on some properties of tridiagonal matrices. As we all know that through orthogonal similarity transformation a real symmetric matrix can be a tridiagonal matrix. So it also needs tridiagonal matrices in symmetric matrix related research.This thesis presents some proofs about the numerical ranges of some tridiagonal matrices. With some conclusions or related theorems, according to the properties of the numerical range of tridiagonal matrix, we prove the numerical ranges of some tridiagonal matrices are elliptical disks and circular disks. A circular disk can be seen as the special elliptical disk, we are able to accurately determine the center of elliptical disk and its length of long axis and minor axis or the center of circular disk and its length of radius.Firstly, for the problem of elliptical disks and circular disks of the numerical ranges of some tridiagonal matrices with zero main diagonal elements, we have the following conclusions.We obtain that the numerical range of the tridiagonal matrix is the elliptical disk, in which a set of standard orthogonal basis defining in Hilbert space satisfies certain conditions. By Toeplitz-Hausdorff theorem we establish the link to the conclusion. We prove that the numerical range of the tridiagonal matrix is circular disk under the condition which the multiplication of the off-main diagonal corresponding elements is zero. We use the mathematical induction. We come to the conclusion based on a series of fundamental theorems such as the Perron-Frobenius theorem etc. We solve the problem that the numerical range of the small size 3?3 matrix which satisfies certain relations between the off-main diagonal elements and their norms is the circular disk.We can exactly have the elliptical standard equation.Secondly, it is proved that the numerical range of the 3?3 tridiagonal matrix which the elements on the main diagonal are an arithmetic progression and elements on off-main diagonal are respectively the same is elliptical disk. The cases are discussed in different situations. Among them, a line segment is seen as a special elliptical disk.Finally, under the condition of elements on the main diagonal we take two different values according to the superscripts of even and odd and off-main diagonal elements which have the same superscript meet somewhat multiple relation. We determine that the numerical range of the tridiagonal matrix is a circular disk.