Some Numerical Characteristic Inequalities For Matrices

Matrix inequality is an important research hotspot in matrix the-ory.In recent decades,matrix inequality has a wide range of ap-plications in quantum information,control theory,image processing,statistics,etc.In this paper,we study some problems on determinantal inequali-ties for sector matrices,unitarily invariant norm inequalities for matri-ces,singular values inequalities of positive semidefinite matrices related to positive linear maps and weighted arithmetic-geometric-harmonic mean inequalities for two accretive matrices.Our main problems are as follows.1.We establish determinantal inequalities for sector matrices by using the relation between the partial traces and the full matrices,the log-concavity of the determinant function over the cone of positive definite matrices.Our results are generalizations of some known in-equalities due to Lin.2.We study unitarily invariant norm inequalities for sector ma-trices.Firstly,we obtain a unitarily invariant norm inequality between accretive-dissipative matrix and its main diagonal blocks by using 2 ×2 block matrix decomposition and triangular inequality.Secondly,we present a Schatten q-norm inequality for sector matrices which gener-alizes Audenaert's result.And then Rotfel'd type inequality involving sector matrices is given.The result can be regarded as an improvement of Zhao and Ni's inequality.Finally,we extend two unitarily invariant norm inequalities for 2 × 2 block positive semidefinite matrices due to Hiroshima to sector matrices.3.We obtain a unitarily invariant norm inequality involving the off-diagonal block of positive partial transpose matrices and the geo-metric mean of its diagonal blocks.At the same time,we give alterna-tive proofs of matrix Holder-type inequalities due to Audenaert,Zou and Jiang,respectively.4.We investigate some singular values inequalities for matrices.Firstly,we extend singular values inequalities for PPT(positive par-tial transpose)matrices to SPT(sectorial partial transpose)matrices.Secondly,we prove the linear map ?:X(?)2tr(X)In-X is 2-PPT by a transparent method.Finally,we establish singular values inequali-ties related to the linea map ? between the off-diagonal blocks and the arithmetic mean of the diagonal blocks which partially solve an open problem raised by Lin.5.We discuss weighted geometric mean equalities for accretive matrices which inherit the same expressions as positive definite matri-ces.And we present several weighted arithmetic-geometric-harmonic mean inequalities for two sector matrices.Our results are generaliza-tions of previous inequalities obtained by researchers.