Further Studies On Some Questions In The Theories Of Metapositive Definite Matrix And Generalized Positive Definite Matrix

In 1970, the notion of unnecessary symmetric positive definite matrix was firstgiven by C.R.Johnson in paper [1] (for (?) 0 ≠ X ∈R^n×1,we have X^T AX>0),and someinequalities for it were obtained in paper [2];In 1984, the notion ofPD ntye of generalizedpositive definite matrix was given by professor TongWenting( (?)D ∈D_n⁺,for (?) 0 ≠ X ∈R^n×1,we have X T DAX>0) , some fundermental properties for it were obtained in paper [5];In1990, the notion of metapositive definite matrix was given by professor TuBoxun( A + ATis symmetric positive definite matrix),and comparatively systematic theories for itwere established in papers [3],[4].In this dissertation ,the fundamental notions and theories in the theory of metapositivedefinite matrix and the theory of generalized positive definite matrix which have beenformed in recent years are presented, some uncompleted theorems in the theory ofmetapositive definite matrix are revised and extended,and the theory of generalizedpositive definite matrix is studied in a deepgoing way,several value results are obtained,therefore the conclusions obtained in this dissertation will bring the theory of metapositivedefinite matrix and the theory of generalized positive definite matrix to completion andextension. The main conclusions obtained in this dissertation are as follows:(1) Therem14(The isolation of generalized Minkowski's inequality for metapositive defini-te matrix)Let A be a non-zero semi-metapositive definite matrix of n order(n ≥ 2),Bsymmetric positive definite matrix of n order. B ?1 A has 2s(02≤s ≤n)complexeigenvalues.1) if s>0, then there exsit a group of positive real numbers μ₁ …μ_n-s,satisfy μ₁… μ₂…μ_n-s=1,such thatand the equality holds if and only if the complex eigenvalues of B^-1A are all equal to± bi , i=?1,b>0(b=const).2) if s=0, then there exsit a group of positive real numbers μ1 ,,μn,satisfy μ1 ? μ2???μn=1,such thatinnnnnniA Bn A1B11A1B111+ ≥∏( +)≥+=μand the equality holds if and only if the real eigenvalues of B-1A are all equal .(2) Therem 15(Kyfan tye of inequality) Let A be a nonzero k-local completely symm-etric semi-metapositive definite matrix of n order(n≥ 2),B symmetric positive definitematrix of n order.and suppose that^,1212A = ????AA Tk AA????B = ????BB1 Tk2 BB¹2 ????,where Am , Bm(1 ≤ m≤k)are the principal submatrices of Ak andBk in the same position,respectively. Am is a singular matrix, α ≥ n ?2m,then the inequalityααα( A + B)/(A+B)m ≥A/Am+B/Bmholds.(3) Therem17(Generalized Hadamard's inequality for local metapositive definite matrix)Let A = ( aij)n×n, a1 1 >0,Shur complement A /( a11)of ( a1 1)in A be a metapositivedefinite matrix.Moreover if1) a ik akj= ajkaki, i,j>k,k=1,2,,n?1.2) a ik aki≥ 0, i>k,k=1,2,,n?1.Then the inequalityA ≤ a11 a22???annholds. and the equality holds if and only if a ik aki= 0, i>k,k=1,2,,n?1.(4) Therem 31(Generalized Oppenheim's inequality for+PD ntye of generalized positivedefinite matrix)Let A be a symmetric positive definite matrix of n order, B= ( )bi jn× nPD n+tye ofgeneralized positive definite matrix ( then there exsists D ∈ Dn+,such that DB is a(semi)metapositive definite matrix). Then the inequalitynnTTTA B≥ AB+D2?1 BD+AB?D2?1BD≥AB+D2?1BD≥A?b11???bholds.(5)The properties of eigenvalue distributions, the structure theorems of submatrix, thenecessary and sufficient conditions which Hadamard (Kronecker) products are generalizedpositive definite matrices for generalized positive definite matrices.