LAUSR

Abstract For a matrix A, let A † denote its Moore–Penrose inverse. A matrix M is called a multiplicative perturbation of T ∈ C m × n if M = E T F ⁎ for some E ∈ C m × m and F ∈ C n × n . Based on the alternative expression for M as M = ( E T T † ) ⋅ T ⋅ ( F T † T ) ⁎ , the generalized triple reverse order law for the Moore–Penrose inverse is obtained as M † = ( ( F T † T ) ⁎ ) † ⋅ ( Y Y † T Z Z † ) L R − 1 † ⋅ ( E T T † ) † , where ( Y Y † T Z Z † ) L R − 1 † is the weighted Moore–Penrose inverse for certain matrices Y , Z , L and R associated to the triple ( T , E , F ) . Furthermore, it is proved that this weighted Moore–Penrose inverse in the resulting expression for M † can be really replaced with T † if ( E T T † ) † E T T † ⋅ T = T ⋅ ( F T † T ) † ( F T † T ) . In the special case that rank ( M ) = rank ( T ) or M is a weak perturbation of T, a simplified version of M † , as well as M M † and M † M , is also derived.