Page 56 - Textos de Matemática Vol. 44

46 M. C. GOUVEIA
Definition 1.1. Let A ∈ Cm×n, and define the linear transformation χ− :Cm →Cn byχ−x=0ifx∈R(A)⊥ andχ−x=(χ|R(A∗))−1xifx∈R(A). The matrix X of χ−, denoted by A−, is called a generalized inverse of A.
ItiseasytocheckthatAA−x=0ifx∈R(A)⊥ andAA−x=xifx∈R(A). Similarly,A−Ax=0ifx∈N(A)=R(A∗)⊥ andA−Ax=xifx∈R(A∗)= R(A−). Thus AA− is the orthogonal projector of Cm onto R(A) and A−A is the orthogonal projector of Cn onto R(A∗) = R(A−). This is the motivation for the definition due to Moore (1935).
Definition 1.2. If A ∈ Cm×n, then a generalized inverse of A is defined to be the unique matrix X such that
(i) AX = PR(A)
(ii) XA = PR(X) = PR(A∗).
An equivalent definition was given by Penrose in (1955).
Definition 1.3. If A ∈ Cm×n, then the unique matrix X in Cn×m that is a solution of the equations
(i) AXA=A
(ii) XAX=X
(iii) (AX)∗ = AX
(iv) (XA)∗ = XA.
is a generalized inverse of A called the Moore-Penrose inverse of A and denoted by A†.
If X is a solution of the ith equation then X is called an (i)-inverse of A. An obvious definition is given for the (i, j)-inverse and (i, j, k)-inverse, i = 1, 2, 3, 4. When i = 1, X is mostly denoted by A(1).
We remark that one situation in which a generalized inverse behaves like a regular inverse, that is, a necessary and sufficient condition for BA−A = B is that R(B∗) ⊂ R(A∗). Similarly for B = AA−B.
If H is infinite dimensional, the definition of the Moore-Penrose inverse for bounded operators with closed range is still meaningful and defines a linear transformation in H characterized as follows.
Definition 1.4. If T ∈ B(H), and R(T) is closed then T† is defined as the unique solution X of the following equivalent systems
(i) TXT=T, XTX=X, (TX)∗=TX, (XT)∗=XT.
(ii) TX = PR(T), N(X∗) = N(T).
(iii) XT = PR(T∗), N(X) = N(T∗)