Further Study Of Generalized Positive Definite Matrices

In1990,the notion of metapositive definite matrix was given by professor TuBoxun and comparatively systematic theories for it were established in papers[1]. Many famous theorem of symmetric positive definite matrix promotion up to the definite matrices. Then Tong Wenting in paper [2], Xia Chang-Fu in paper [3], Yang Shichun, NGO van Quyen in paper [4] to promote subpositive definite matrices for the generalized positive definite matrix. That isPD,Ps+ and PI+ the generalized positive definite matrices. This article focuses on some properties of PD,Ps+and PI+. And predecessors on the basis of studies to further promote the concept of generalized positive definite matrices. let A∈mn×n be given,0≠X=(x1,x2,···,xn)T∈Rn×1, if(?), D1∈PD for XT D1AX>0,then A is further generalized positive definite matrices for A∈PD+.Some properties and to verify whether there are similar and conclusion. This article gives the following conclusions: Theorem 3.1. if A∈PD+, there exists Positive diagonal matrix D and D2∈P1.then①D-1D2A∈P1②D2DA∈P1. Proposition 3.1.1 if A∈PD+, there exists Positive diagonal matrix D and S1,S2∈P1, then A=S1DS2. Proposition 3.3.2.if A∈PD+, then AT∈PD+. Proposition 3.3.3.if A∈PD+, then A-1∈PD+ Theorem 3.2.ifA∈PD+,A(?)is principal submatrix of A, then A(?)∈PD...