Linear operators preserving inverses of matrices
BU Chang-jiang · Harbin Gongcheng Daxue Xuebao/Journal of Harbin Engineering University · 2005
One of the problems of invariance preservation on matrix space, preserving linear operator so that the invariant is matrix, is studied. Eliminating the restriction of the characteristic of field,the forms of invertible linear operator preserving inverses of matrices over full matrix space with at least four elements are described. Using the conclusions of idempotent-preserving, it is proved that f is the invertible linear operator from W-n(F) into M-n(F) preserving inverses of matrices, if and only if there exists P∈GL-n(F), such that f(A)=ePA~TP~(-1),A∈M-n(F),e=±1∈F,or there exists P∈GL-n(F),such that f(A)=eAP~T,A∈M-n(F),e=±1∈F.