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THE GENERALIZED INVERSE ${A_{T,*}}^{(2)}$ AND ITS APPLICATIONS
Cao, Chong-Guang,Zhang, Xian 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.11 No.1
The existence and representations of some generalized inverses, including ${A_{T,*}}^{(2)},\;{A_{T,*}}^{(1,2)},\;{A_{T,*}}^{(2,3)},\;{A_{*,S}}^{(2)},\;{A_{*,S}}^{(1,2)}\;and\;{A_{*,S}}^{(2,4)}$, are showed. As applications, the perturbation theory for the generalized inverse {A_{T,S}}^{(2)} and the perturbation bound for unique solution of the general restricted system $A_{x}$ = b(dim(AT)=dimT, $b{\in}AT$ and $x{\in}T$) are studied. Moreover, a characterization and representation of the generalized inverse ${A_{T,*}}^{(2)}$ is obtained.
INVOLUTION-PRESERVING MAPS WITHOUT THE LINEARITY ASSUMPTION AND ITS APPLICATION
Xu, Jin-Li,Cao, Chong-Guang,Wu, Hai-Yan The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.1
Suppose F is a field of characteristic not 2 and $F\;{\neq}\;Z_3$. Let $M_n(F)$ be the linear space of all $n{\times}n$ matrices over F, and let ${\Gamma}_n(F)$ be the subset of $M_n(F)$ consisting of all $n{\times}n$ involutory matrices. We denote by ${\Phi}_n(F)$ the set of all maps from $M_n(F)$ to itself satisfying A - ${\lambda}B{\in}{\Gamma}_n(F)$ if and only if ${\phi}(A)$ - ${\lambda}{\phi}(B){\in}{\Gamma}_n(F)$ for every A, $B{\in}M_n(F)$ and ${\lambda}{\in}F$. It was showed that ${\phi}{\in}{\Phi}_n(F)$ if and only if there exist an invertible matrix $P{\in}M_n(F)$ and an involutory element ${\varepsilon}$ such that either ${\phi}(A)={\varepsilon}PAP^{-1}$ for every $A{\in}M_n(F)$ or ${\phi}(A)={\varepsilon}PA^{T}P^{-1}$ for every $A{\in}M_n(F)$. As an application, the maps preserving inverses of matrces also are characterized.
A MATRIX INEQUALITY ON SCHUR COMPLEMENTS
YANG, ZHONG-PENG,CAO, CHONG-GUANG,ZHANG, XIAN 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1
We investigate a matrix inequality on Schur complements defined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements defined respectively by inverses and Moore-Penrose generalized inverses (see Wang et al. [Lin. Alg. Appl., 302-303(1999)163-172] and Liu and Wang [Lin. Alg. Appl., 293(1999)233-241]). Moreover, the non-uniqueness of $\{1\}$-generalized inverses yields the complicatedness of the extension.
Involution-preserving maps without the linearity assumption and its application
Jin-li Xu,Chong-guang Cao,Hai-yan Wu 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.1
Suppose F is a field of characteristic not 2 and F ≠ Z3. Let Mn(F) be the linear space of all n×n matrices over F, and let Γn(F) be the subset of Mn(F) consisting of all n×n involutory matrices. We denote by Φn(F) the set of all maps from Mn(F) to itself satisfying A−λB∈ Γn(F) if and only if ∅(A)− λ∅(B)∈ Γn(F) for every A,B ∈ Mn(F) and λ ∈ F. It was showed that ∅∈Φn(F) if and only if there exist an invertible matrix P ∈ Mn(F) and an involutory element ε such that either ∅(A) = εPAP−1 for every A ∈ Mn(F) or ∅(A) = εPAT P−1 for every A ∈ Mn(F). As an application, the maps preserving inverses of matrces also are characterized. Suppose F is a field of characteristic not 2 and F ≠ Z3. Let Mn(F) be the linear space of all n×n matrices over F, and let Γn(F) be the subset of Mn(F) consisting of all n×n involutory matrices. We denote by Φn(F) the set of all maps from Mn(F) to itself satisfying A−λB∈ Γn(F) if and only if ∅(A)− λ∅(B)∈ Γn(F) for every A,B ∈ Mn(F) and λ ∈ F. It was showed that ∅∈Φn(F) if and only if there exist an invertible matrix P ∈ Mn(F) and an involutory element ε such that either ∅(A) = εPAP−1 for every A ∈ Mn(F) or ∅(A) = εPAT P−1 for every A ∈ Mn(F). As an application, the maps preserving inverses of matrces also are characterized.
LINEAR MAPS PRESERVING PAIRS OF HERMITIAN MATRICES ON WHICH THE RANK IS ADDITIVE AND APPLICATIONS
TANG, XIAO-MIN,CAO, CHONG-GUANG 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.19 No.1
Denote the set of n ${\times}$ n complex Hermitian matrices by Hn. A pair of n ${\times}$ n Hermitian matrices (A, B) is said to be rank-additive if rank (A+B) = rank A+rank B. We characterize the linear maps from Hn into itself that preserve the set of rank-additive pairs. As applications, the linear preservers of adjoint matrix on Hn and the Jordan homomorphisms of Hn are also given. The analogous problems on the skew Hermitian matrix space are considered.
Additive operators preserving rank-additivity on symmetry matrix spaces
Xiao-Min Tang,Chong-Guang Cao 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.14 No.-
We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. Let Sn(F) be the space of all n × n symmetry matrices over a field F with 2, 3 2 F, then T is an additive injective operator preserving rank-additivity on Sn(F) if and only if there exists an invertible matrix U 2 Mn(F) and an injective field homomorphism of F to itself such that T(X) = cUXUT , 8X = (xij ) 2 Sn(F) where c 2 F,X = ((xij )). As applications, we determine the additive operators preserving minus-order on Sn(F) over the field F.
A matrix inequality on Schur complements
Zhong-peng Yang,Chong-guang Cao,Xian Zhang 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1-2
We investigate a matrix inequality on Schur complements de- fined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements de- fined respectively by inverses and Moore-Penrose generalized inverses (see Wang et al. [Lin. Alg. Appl., 302-303(1999)163-172] and Liu and Wang [Lin. Alg. Appl., 293(1999)233-241]). Moreover, the non-uniqueness of {1}-generalized inverses yields the complicatedness of the extension.
Linear maps preserving pairs of Hermitian matrices on which the rank is additive and applications
Xiao-Min Tang,Chong-Guang Cao 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.19 No.1-2
Denote the set of n × n complex Hermitian matrices by Hn. A pair of n × n Hermitian matrices (A,B) is said to be rank-additive if rank (A+B) = rank A+rank B. We characterize the linear maps from Hn into itself that preserve the set of rank-additive pairs. As applications, the linear preservers of adjoint matrix on Hn and the Jordan homomorphisms of Hn are also given. The analogous problems on the skew Hermitian matrix space are considered.