Representations And Perturbations Analysis For The Moore-Penrose Inverses

The multiplicative perturbation of matrices has important applications in various fields such as the structured weighted least squares problems, representations for the Moore-Penrose inverse of certain partitioned matrices and so on. A multiplicative perturbation of T ∈ Cm×n has the form M= ETF*,where E ∈ Cm×m and F∈Cn×n. Let T(?) and M(?) be the Moore-Penrose inverses of T and M, respectively. Much effort has been paid in the study of the representation for the M(?) in terms of E, T, F and their Moore-Penrose inverses, and almost all of the literatures are focused on the case that both E and F are nonsingular. Yet, little has been done in the case that either E or F is singular, which is the concern of this dissertation. For the triple (E, T, F), two matrices LE and RF defined as LE= ETT(?)+Im-TT(?), RF= T(?)TF+In-T(?)T are introduced. As a result, the multiplicative perturbation M= ETF* turns out to be LET(RF)*. In view of this new expression of M, two new types of multiplicative perturbations called strong perturbation and weak perturbation are introduced, and a formula for M* is derived in the general case that M is only a weak perturbation of T. Based on this formula, upper bounds for ||M(?)-T(?)||2 are derived. The sharpness of the obtained upper bound is illustrated by some numerical examples.The weighted Moore-Penrose inverse AMN(?) has important application in the study of the weighted least squares problems. Up to now, most literatures are focused on the case that both M and N are fixed, whereas A is variable. Little has been done in the opposite case that A is fixed, whereas both M and N are variable. For a complex matrix A∈Cm×n, the relationship between the weighted Moore-Penrose inverse AM1N1(?) and AM2N2(?) is studied in this dissertation, and an important formula is derived. Based on this formula, this dissertation initiates the study of the perturbation estimations for AMN(?) in the case that A is fixed, whereas both M and N are variable. The obtained norm upper bounds are then applied to the perturbation estimations for the solutions to the weighted linear least squares problems.