Representations and norm estimations for the Moore-Penrose inverse of multiplicative perturbations of matrices

A multiplicative perturbation of a matrix T is an element of C-mxn has the form ETF* with E is an element of C-mxm and F is an element of C-nxn, which can be expressed alternatively as LET(R-F)*, where T+ is the Moore-Penrose inverse of T, L-E and R-F are introduced as L-E = ETT+ + I-m -TT+, R-F = T+ T-F + I-n -T+ T. In view of the above L-E and R-F, a new type of multiplicative perturbation called weak perturbation is introduced. A formula for (ETF*)+ is derived in the general case that ETF* is a weak perturbation of T. Based on this formula, an upper bound for parallel to(ETF*)(+) -T+parallel to 2 is derived. The sharpness of the obtained upper bound is illustrated by some numerical examples.