Study On Some Matrices Problems Over Real Quaternion Division Algebra

Quaternion is another new number system after the complex. The quaternion field is an extension of complex field. It is a relative independent system and has great difference with the complex field because of the non- commutative multiplication of quaternions.In recent years, the algebra problems over quaternion division algebra have drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalues, and system of quaternion matrices equations and so on.However, there are many questions to be studied in quaternion algebra, such as the estimation of quaternion matrices eigenvalues, diagonalization, the distribution for the generalized eigenvalues of quaternion matrices, the perturbation of solutions of quaternion right linear equations, the sub- positive of quaternion matrices and the solutions of quaternion matrix equations, e.t.c.The purpose of this paper is to discuss some important algebra properties on quaternion division algebra. The main results and innovations are listed as the following:1. The famous Gerschgorin's disk theorem over complex field is generalized to real quaternion field by the definition of eigenvalues. Two different forms of distribution of the left eigenvalues and right eigenvalues are obtained and some properties of eigenvalues of quaternion diagonally dominant matrices are also discussed. Moreover, the definition of generalized eigenvalues and its properties are given, and then we get a conclusion that the generalized eigenvalue of regular matrix is real. The estimation of the upper and lower bound of quaternion matrices'generalized eigenvalues is obtained with the guidance of the generalized Rayleigh quotient.2. The necessary and sufficient condition for the diagonalized quaternion matrices is given, which is different from real (complex) field. Meanwhile, the difference between diagonalization of complex matrices and quaternion matrices is shown.3. The estimation of spectral radius is discussed based on its concept.4. We solve the quaternion matrices inversion, the estimation of linear equations error and eigenvalues according to quaternion vector and matrices norm theory.5. We also study on the metapositive definite matrices over quaternion division algebra and some results are obtained.6. The Kronecker product is important. The quaternion field is different from the complex field because of the non- commutative multiplication of quaternions. Then we study the solutions of quaternion Lyapunov matrix equations and the Stein Equations by Kronecker product over quaternion division algebra.