qLet AÎ Fm´n. XÎ Fn´m is a right inverse if AX = I. One of the right inverses is given by X=A*(AA*)-1. YA = I
then Y is a left inverse of A.
qPseudo-inverse or Moore-Penrose
inverse A+:
•(i) A A+A=A; (ii) A+A A+= A+; (iii) (A A+)*=A A+; (iv) (A+A)*=A+A.
• Pseudo-inverse is
unique.
• Let A=BC where B has a full column rank
and C has full row rank. Then A+=C*(CC*)-1(B*B)-1 B* .
• or let A=USV* with
and Sr=diag{s1, s2, … ,sr}.
•
• Then A+=VS+U* with