Study On The Numerical Characteristics Of Matrices And The Positive Definite Matrices

Matrix computations and matrix analysis have wide application in economics, computational mathematics, computer graphics and image analysis, and control theory etc. The purpose of this thesis is to discuss the numerical characteristics of matrices and the properties and discrimination of the positive definite matrices. The main results and innovations are as follows:1. For arbitrarily complex matrix ( )M∈M nC,it can be written in the following form where Ak×kis k th sequential principal matrix of M . For convenience, A , B , C ,and , respectively . If then M is said to be a TD matrix. We denote the set of TD matrix by TDn . In section two, we prove that all the eigenvalues of M which is a TD matrix are located in one disk. The estimation is always more convenient than the earlier ones.2. For arbitrarily complex matrix, we prove that all the eigenvalues of M are located in the following disk After that we present a sufficient condition of that Linear Time-Invariant System is asymptotically stable in equilibrium position. We shall conclude this section with some numerical examples.3. Based on linear algebra theories, two lower bounds for the smallest singular value are obtained. The first lower bound is simple and suitable for computer implementation. The second lower bound is better than those the earlier for some matrices.4. We obtain several results which are about symmetric product, estimation for real part, spectral radius, and determinant of positive definite matrices. These are the improvement of the earlier results. At the same time, we present that the necessary and sufficient condition for Hermitian matrices being positive definite matrices is that Hermitian matrices'real representation matrices are real symmetric positive definite matrices. We shall conclude this section with the algorithm based on optimization theory for determining positive definiteness of matrices and the numerical examples.5. As is well-known that the main obstacle in the study of quaternion division algebra and quaternion matrices is the non-commutative multiplication of quaternion, and there are some difficulties in working on quaternion matrix challenges. Based on the properties of the right eigenvalues of quaternion matrices and some inequalities, we present some easily computed inclusion regions in 4D spaces that are guaranteed to include the right eigenvalues of quaternion matrices.