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2009•Journal of Natural Science of Heilongjiang UniversityRequires access

Group inverses of block matrices with a sub-block in the form of sum or difference over skew fields

Xiaoguang Han

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Abstract

For an n×n matrix A over a skew field,the n×n matrix X is called the group inverse of A,if it satisfying the matrix equations AXA=A,XAX=X,AX=XA.Investigating the existence and representation of group inverses for block matrices not only have important theoretical significance but also extensive application value.At present,the existence and representation of group inverse for block matrix(CAB0)is still an open problem.Necessary and sufficient conditions for existence and representation of group inverse of the block matrix(CAB0)over skew fields is given for the cases:(i)A#,B# exist and C=±(A+B);and(ii)A#,B# exist and C=±(A-B)).

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What this paper is about

For an n×n matrix A over a skew field,the n×n matrix X is called the group inverse of A,if it satisfying the matrix equations AXA=A,XAX=X,AX=XA.Investigating the existence and representation of group inverses for block matrices not only have important theoretical significance but also extensive application value.At present,the existence and representation of group inverse for block matrix(CAB0)is still an open problem.Necessary and sufficient conditions for existence and representation of group inverse of the block matrix(CAB0)over skew fields is given for the cases:(i)A#,B# exist and C=±(A+B);and(ii)A#,B# exist and C=±(A-B)).

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Available abstract

For an n×n matrix A over a skew field,the n×n matrix X is called the group inverse of A,if it satisfying the matrix equations AXA=A,XAX=X,AX=XA.Investigating the existence and representation of group inverses for block matrices not only have important theoretical significance but also extensive application value.At present,the existence and representation of group inverse for block matrix(CAB0)is still an open problem.Necessary and sufficient conditions for existence and representation of group inverse of the block matrix(CAB0)over skew fields is given for the cases:(i)A#,B# exist and C=±(A+B);and(ii)A#,B# exist and C=±(A-B)).

Key concepts: Mathematics, Block (permutation group theory), Inverse, Skew, Group (periodic table), Matrix (chemical analysis), Combinatorics, Block matrix

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