Rearrangement Inequality Of Self - Conjugate Matrices

Self-conjugate matrices are a kind of special matrices of quaternion. They can be seen more extensive and more generic matrices, which include symmetric matrices and hermitian matrices. In this paper, according to these properties, the Hardy-Littlewood-P(?)lya rearrangement inequality is extended to self-conjugate matrices. We first give the content and proof of the Hardy-Littlewood-P(?)lya rearrangement inequality, as well as the Hardy-Littlewood-P(?)lya rearrangement inequality about hermitian matrices; Then we introduce the important definitions of self-conjugate matrices and some basic theorems and properties of numerical characteristic;Finally, we do some researches such that the Hardy-Littlewood-P(?)lya rearrangement inequality is extended to self-conjugate matrices with respect to determinant, trace, kronecker product and hadamard product.The thesis is composed of three chapters.The chapter 1 is the introduction, which gives the Hardy-Littlewood-P(?)lya rearrangement inequality of real numbers and some mathematical signs.In chapter 2, we introduce the definitions of the self-conjugate matrices, quaternion(semi-)positive matrices, trace, kronecker product and hadamard product. Then, we research some properties of positive, eigenvalue, trace and the operation rules of kronecker product and hadamard product.In chapter 3, taking advantage of the important properties of self-conjugate matrices, the Hardy-Littlewood-P(?)lya rearrangement inequality can be extended to the self-conjugate matrices such that we can study commutative self-conjugate matrices rearrangement inequality, as well as the rearrangement inequality of determinant, trace, kronecker product and hadamard product.