Research On Generalized Positive Definite Matrices

In this dissertation,we systematically discuss the fundamental properties and the necessary and sufficient conditions of real symmetric positive definite matrices,Hermitian positive definite matrices and two types of generalized positive definite matrices-metapositive definite matrices and complex positive definite matrices.Finally,we discuss the relation among these four types of positive definite matrices.In Chapter 1,we give some other important properties other than introducing some important ones about Hermitian matrices ,Hermitian positive definite matrices and real symmetric positive definite matrices.Especially,we summarize and establish some of their necessary and sufficient conditions roundly.In Chapter 2,we give some important properties of metapositive definite matrices,the expression of real and imaginary part about eigenvalues of real square matrices as well as the range of eigenvalues of real square matrices(especially the range of their imaginary part ).On the other hand,we sum up,establish and extend some necessary and sufficient conditions of metapositive definite matrices roundly.In Chapter 3,we give some important properties of complex positive definite matrices,the expression of real and imaginary part about eigenvalues of complex square matrices as well as the range of eigenvalues of complex square matrices.On the other hand,we sum up,establish and extend some necessary and sufficient conditions of complex positive definite matrices roundly.In Summary, we discuss in terms of set theory the relation among real symmetric positive definite matrices,Hermitian positive definite matrices and two types of generalized positive definite matrices-metapositive definite matrices and complex positive definite matrices .