We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between <em>C</em>*-...We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between <em>C</em>*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.展开更多
In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Mo...In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.展开更多
The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner t...The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.展开更多
This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invar...This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.展开更多
For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(...For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A⊕B)+φc(A,B))≤sj(A+|B1/2A1/2|)⊕(B+|A1/2B1/2|),where sj(X)denotes the j-th largest singular value of X andφc(A,B):=1/2((1+c)|B1/2A1/2|(1-c)A1/2B1/2(1-c)B1/2A1/2(1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.展开更多
文摘We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between <em>C</em>*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.
文摘In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.
基金supported by the research grant UL020/08-Y2/MAT/ JXQ01/FST from University of Macao
文摘The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.
基金Supported by the Scientific Research Project of Chongqing Three Gorges University(11QN-21)
文摘This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.
基金Supported by the Natural Science Foundation of Anhui Province(1708085QA05)the Natural Science Foundation of Anhui Higher Education Institutions of China(KJ2019A0588,KJ2020ZD008)。
文摘For positive real numbers a,b,a+b≤max{a+b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A⊕B)+φc(A,B))≤sj(A+|B1/2A1/2|)⊕(B+|A1/2B1/2|),where sj(X)denotes the j-th largest singular value of X andφc(A,B):=1/2((1+c)|B1/2A1/2|(1-c)A1/2B1/2(1-c)B1/2A1/2(1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.
