Estimation And Location Analysis Of Characteristic Quantities Of Some Special Matrices

Matrix theory is the main content of linear algebra, while the estimation of matrix eigenvalue is the main research point of matrix theory. Theoretically, for any given complex matrix, all of its characteristic quantities, such as its spectrum, spectral radius, spread, trace and determinant, condition number, are fixed constants. But it is well known that it is difficult to obtain any characteristic quantity for a high order matrix or system. And in many cases, it is not necessary to get the exact value of its characteristic quantity, only need to determine its range. For example, under certain conditions, if one can establish a concrete interval(a disk, an elliptic, or a rectangle) included a given matrix’s eigenvalues, then one can conclude that its spread is less than the length of interval(the diameter of disk, the major axis of elliptic or the length of the diagonal line of rectangle). The most important subjects is the spread of a matrix in this paper, its mainly used to depict the distribution of eigenvalues denseness, so it is an important and unique algebraic characteristics.In this article, our purpose is to discuss the research progress of the estimation and location of matrix eigenvalues and determine the upper and lower bounds of matrix spread. We will also discuss the eigenvalues and spread of some special matrices.The mainly research contents contain:(1) Using the characteristics of Cyclic matrix, we obtained eigenvalues of eachCyclic matrix that are all located in a disc:And its eigenvalues are all presented in the following set:(2) Placing the results of the literature [4] and [9] for Real symmetric matrices, two more accurate lower bound of spread of Real symmetric matrices were obtained.(3) By means of the general matrix’s spread, and combining the characteristics of Toeplitz matrices, Hankel matrices and Cycle matrices, the upper bound of the spread of these three types of special matrices were given.Matrix’s eigenvalue is an important research field in matrix algebra. It has a more important role in the study of the matrix theory.