The Preserver Problems About M-P Inverses Of Matrix Spaces

Preserving Problems of matrix spaces is an active research subject in Matrix The-ory which concern the characterization of maps between the spaces of matrices that pre-serve some invariants. Since the Generalized Inverses of matrices have wide applicationsin many areas such as differential equations statics optimal theory so it is still active asa study subject. M-P inverse is an important Generalized Inverses of a matrix the mainpurpose of this paper is to investigate the maps preserving M-P inverses of matrices.One of important techniques in the study of Preserver Problems is to reduce newPreserver Problems to the known ones such as idempotence rankone preserver and soon. In terms of the particularity and complication of M-P inverses of matrices reducingthe linear maps preserving M-P inverses of matrices to the idempotent preserver is moreor less difficulty. In this paper We study the problem by searching some particularmatrices directly.In the second part of this paper Suppose R is a commutative PID of characteristic2 with a unit u other than 1 such that u~2 = 1 u~3 = 1. We denote by Mn(R) andS_n(R) the spaces of n×n full matrices and symmetric matrices over R respectively.By searching some particular matrices using the method of characterizing the imagesof the bases the invertible linear maps from S_n(R) to S_n(R) preserving M-P inversesof matrices are characterized. When R is a commutative PID with a unit 2 the linearinjections from S_n(R) to Mn(R) preserving M-P inverses of matrices are characterizedalso. Suppose R is a commutative integral domain with a unit u other than 1 such thatu~2 = 1 u~3 = 1. We denote by Tn(R) the space of n×n upper triangular matrices overR. In the third part the linear maps from Tn(R) to Tn(R) preserving M-P inverses ofmatrices are characterized.