On The Perturbation Of Weighted Drazin Inverse And Weighted Group Inverse Of Rectangular Matrices

Cen (Math. Numer. Sin.29(1):39-48,2007) defined a weighted group inverse of rectangular matrices and gave some necessary and sufficient conditions for existence and some expressions of the weighted group inverse of rectangular matrices by the universal factorization. For given matrices A∈Cm×n and W∈Cn×m , ifΧ∈Cm×n satisfies (W1) AWXWA= A,(W2) XWAWX=X, (W3) AWX=WA then X is called the W—weighted group inverse, which is denoted by Aw#. Paper [2] added some new existence conditions and new expressions, more importantly, the author also gave an existence condition and several expressions for the weighted group inverse defined by Cline and Greville which made a connection between the two weighted inverses. In this paper, for given rectangular matrices A and E which are m x n matrices and B= A+E, we discuss the analytical perturbation of the weighted group inverse Aw* and give the upper bounds for||Bw#||.Castro-Gonzalez, Robles, and Velez-Cerrada (SIAM J. Matrix Anal. Appl.,30 (2008), pp.882-897) defined a condition (Cs) for a singular square matrix in the perturbation analysis of the Drazin inverse. Similarly, the condition (Cs) of a rect-angular matrix also exists. The perturbation is said to be a stable perturbation if the condition is satisfied. In this paper, only under the condition that B is a sta-ble perturbation of A, an explicit formula for the W-weighted Drazin inverse Ba,w is provided and some upper bounds for||Bd,w-Ad,w||/||Ad,w||. are derived under certain conditions. Furthermore, as a special weighted Drazin inverse, the stable perturbation of the weighted group inverse can also be analyzed.