The Group Inverse Of Some Block Matrices And Linear Maps Preserving Annihilating Polynomial

This paper outlines the study of preserving problems and the significance of the generalized inverses of matrices and the research status. We also gives the basic knowledge of generalized inverses of matrices and matrix function.Generalized inverses of matrices are an active research area in matrix theory. In 1979,Campbell and Meyer proposed the problem to find an representation forthe Drazin and group inverse of any 2×2 block matrix(?).But so far,it has not yet resolved. Some articles are in certain special circumstances for research. In this paper,we study the existences and representations for the groupinverse of twenty-seven block matrices of the form(?).Let A,B,C be thematrices from the set {P, PP~*,PP~*P},Suppose that P is a given idempotent matrix over the field of complex numbers, P~* is the transpose conjugate of P.Preserving invariant problem over matrix space is an active research area. A lot of scholars are interested in studying the matrices algebra linear square-zero preservers on spaces of square matrices,preservers of k -power,linear preservers of involutory matrices. The research on these problems all may come down to the problems of preservers annihilating polynomial.Seeking the new invariant to study preserver problem is a driving force which is full of vigor. In the paper the problem of preservers annihilating polynomial is submitted and partly resolved, which is a new invariant study. Let C be a field, M_n(C) be the n×n full matrix algebra over C.We determine the forms of linear maps from M_n(C) to M_n(C) preserving second order annihilating polynomial.