The Moore—Penrose or generalized inverse | Basis (Linear Algebra) | Matrix (Mathematics)

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  One would like to be able to find a matrix (or matrices) C, such that solutions of (1) are of the form Cb. But if b0R(A), then (1) has no solution. This will eventually require us to modify our concept of what a solution of (1) is. However, as the applications will illustrate, this is not as unnatural as it sounds. But for now we retain the standard definition of solution.
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  The Moore—Penrose or generalized inverse 1. Basic definitions Equations of the form Ax = b,AEC lx  ,xeC ,beC m  1 ccur in many pure and applied problems. If AEC ' 1 <' 1  and is invertible, then the system of equations (1) is, in principle, easy to solve. The unique solution is x = A   b. If A is an arbitrary matrix in C ' , then it becomes more difficult to solve (1). There may be none, one, or an infinite number of solutions depending on whether beR(A) and whether n-rank (A) > 0. One would like to be able to find a matrix (or matrices) C, such that solutions of (1) are of the form Cb. But if b0R(A), then (1) has no solution. This will eventually require us to modify our concept of what a solution of (1) is. However, as the applications will illustrate, this is not as unnaturalas it sounds. But for now we retain the standard definition of solution. To motivate our first definition of the generalized inverse, consider the functional equation Y =f(x),xe 9 c III,   2) where f is a real-valued function with domain 9ã One procedure for solving (2) is to restrict the domain off to a smaller set 9' so that  f=   s one to one. Then an inverse function f -' from R(f) to 9 is defined by f - '(y) = x if x e S' and f (x) = y. Thus f - '(y) is a solution of (2) for y e R(f ). This is how the arcsec, arcsin, and other inverse functions are normally defined.The same procedure can be used in trying to solve equation (1). Asusual, we let A be the linear function from C into Cm defined by Ax = Ax for xEC . To make A a one to one linear transformation it must berestricted to a subspace complementary to N(A). An obvious one is N(A) 1 = R(A*). This suggests the following definition of the generalized inverse. Definition 1.1.1 Functional definition of the generalized inverse   f    D  o  w  n   l  o  a   d  e   d   0   4   /   1   9   /   1   6   t  o   1   3   2 .   2   3   9 .   1 .   2   3   1 .   R  e   d   i  s   t  r   i   b  u   t   i  o  n  s  u   b   j  e  c   t   t  o   S   I   A   M    l   i  c  e  n  s  e  o  r  c  o  p  y  r   i  g   h   t  ;  s  e  e   h   t   t  p  :   /   /  w  w  w .  s   i  a  m .  o  r  g   /   j  o  u  r  n  a   l  s   /  o   j  s  a .  p   h  p  THE MOORE -PENROSE OR GENERALIZED INVERSE 9 AEC ' , define the linear transformation At :Cm -+ C by Atx = 0 if xeR(A)- and Atx = (A R   A *) ) 1 x if xeR(A). The matrix of At is denoted At and is called the generalized inverse of A. It is easy to check that AAtx = 0 if xeR(A)l and AAtx = x if xER(A). Similarly, AtAx = 0 if xeN(A) = R(A*) and AtAx = x if xER(A*) = R(At). Thus AAt is the orthogonal projector of C ' onto R(A) while AtA is the orthogonal projector of Cn onto R(A*) = R(At). This suggests a second definition of the generalized inverse due to E. H. Moore Definition 1.1.2 Moore definition of the generalized inverse   f A u C   21  x  , hen the generalized inverse of A is defined to be the unique matrix At such that (a) AAt = P R(A) , and (b) AtA = P R(A , ) . Moore's definition was given in 1935 and then more or less forgotten. This is possibly due to the fact that it was not expressed in the form of Definition 2 but rather in a more cumbersome (no pun intended) notation. An algebraic form of Moore's definition was given in 1955 by Penrose who was apparently unaware of Moore's work. Definition  1. 1.3 Penrose definition of the generalized inverse   f AeC' , then At is the unique matrix in C '2  such that (i) AAtA = A, (ii) AtAAt = At, (iii) (AAt)* = AAt, (iv) (AtA)* = AtA. The first important fact to be established is the equivalence of thedefinitions. Theorem 1.1.1 The functional, Moore and Penrose definitions of the generalized inverse are equivalent. Proof We have already noted that if At satisfies Definition 1, then it satisfies equations (a) and (b). If a matrix At satisfies (a) and (b) then it immediately satisfies (iii) and (iv). Furthermore (i) follows from (a) by observing that AAtA = P R(A) A = A. (ii) will follow from (b) in a similar manner. Since Definition 1 was constructive and the At it constructs satisfies (a), (b) and (i) -(iv), the question of existence in Definitions 2 and 3 is already taken care of. There are then two things remaining to be proven. One is that a solution of equations (i) -(iv) is a solution of (a) and (b). The second is that a solution of (a) and (b) or (i) -(iv) is unique. Suppose then that At is a matrix satisfying (i) -(iv). Multiplying (ii) onthe left by A gives (AAt) 2  = ( AAt). This and (iii) show that AAt is an orthogonal projector. We must show that it has range equal to the range    D  o  w  n   l  o  a   d  e   d   0   4   /   1   9   /   1   6   t  o   1   3   2 .   2   3   9 .   1 .   2   3   1 .   R  e   d   i  s   t  r   i   b  u   t   i  o  n  s  u   b   j  e  c   t   t  o   S   I   A   M    l   i  c  e  n  s  e  o  r  c  o  p  y  r   i  g   h   t  ;  s  e  e   h   t   t  p  :   /   /  w  w  w .  s   i  a  m .  o  r  g   /   j  o  u  r  n  a   l  s   /  o   j  s  a .  p   h  p  10 GENERALIZED INVERSES OF LINEAR TRANSFORMATIONS of A. Using (i) and the fact that R(BC) c R(B) for matrices B and C, we get R(A) = R(AAtA) 9 R(AAt) c R(A), so that R(A) = R(AAt). Thus AAt = P R(A)  as desired. The proof that AtA = PR(A') is similar and is left to the reader as an exercise. One way to show uniqueness is to show that if At satisfies (a) and (b), or (i) —(iv), then it satisfies Definition 1. Suppose then that At is a matrix satisfying (i) —(iv), (a), and (b). If xeR(A) 1 , then by (a), AAtx = 0. Thus by (ii) Atx = AtAAtx = At0 = 0. If xeR(A),then there exist yeR(A*) such that Ay = x. But Atx = AtAy = y. The last equality follows byobserving that taking the adjoint of both sides of (i) gives P R(At) A = A* so that R(A*) 9 R(At). But y = (AI R(A*) ) - 'x. Thus At satisfies Definition 1. As this proof illustrates, equations (i) and (ii) are, in effect, cancellation laws. While we cannot say that AB = AC implies B = C, we can say that if ATAB = AtAC then AB = AC. This type of cancellation will frequently appear in proofs and the exercises.For obvious reasons, the generalized inverse is often referred to as theMoore—Penrose inverse. Note also that if A e C and A is invertible,then A -1  = At so that the generalized inverse lives up to its name. 2. Basic properties of the generalized inverse Before proceeding to establish some of what is true about generalized inverses, the reader should be warned about certain things that are not true. While it is true that R(A*) = R(At), if At is the generalized inverse, condition (b) in Definition 2 cannot be replaced by AtA = P R(A ,^. Example 1.2.1 Let A = [  0 , X = 1   ,  01. Since XA = AX =   1  01   P _ P   satisfies AX = P R(A)  and XA = P   ut O O J   (A)   (A')^   (Aã)' At  = [1 /2  0] and hence X # At. Note that XA * P R(X) and thus XAX * X. If XA = P R(A*) , AX = P R(A) , and in addition XAX = X, thenX = At. The proof of this last statement is left to the exercises. In computations involving inverses one frequently uses (AB) - ' _ B - 'A - ' if A and B are invertible. This fails to hold for generalized inverses even if AB = BA. Fact 1 2 1 If A E C ` , B c C p, then (AB)t is not necessarily the same as BtAt. Furthermore (At) 2  is not necessarily equal to (AZ)t. Example 1.2.2 Let A = [   ]. Then At = 2 [  1 0 J . Now    D  o  w  n   l  o  a   d  e   d   0   4   /   1   9   /   1   6   t  o   1   3   2 .   2   3   9 .   1 .   2   3   1 .   R  e   d   i  s   t  r   i   b  u   t   i  o  n  s  u   b   j  e  c   t   t  o   S   I   A   M    l   i  c  e  n  s  e  o  r  c  o  p  y  r   i  g   h   t  ;  s  e  e   h   t   t  p  :   /   /  w  w  w .  s   i  a  m .  o  r  g   /   j  o  u  r  n  a   l  s   /  o   j  s  a .  p   h  p  THE MOORE —PENROSE OR GENERALIZED INVERSE 11 A Z  = A while Ate = ?At. Thus (At) 2 A 2  = ? AtA which is not a projection.Thus (At)2 * ( A')t . Ways of calculating At will be given shortly. The generalized inverses in Examples 1 and 2 can be found directly from Definition 1 without too much difficulty. Examples 2 illustrates another way in which the properties of the generalized inverse differ from those of the inverse. If A is invertible, then 2 ea(A) if and only if 1  ea(A -1 ). If A = [ ß   I as in Example 2, then 6(A) =  {1,0} while a(At) = { i , 0 } If A is similar to a matrix C, then A and C have the same eigenvalues, the same Jordan form, and the same characteristic polynomial. None of these are preserved by taking of the generalized inverse. 1 1 —1 Example 1 2 2 Let A =   0   2 . Then A = BJB   where —1 1   1 0 1   1 0 B= 01 1 nd J= 10 00 The characteristic polynomial of A 1 0  oj   0 2 000 and J is 2 2 (2 — 2) with elementary divisors 2 2  and I — 2. Jt = 1 00 0 0 1/2 and the characteristic polynomial of Jt is 2 2 (1 — 1/2) with elementary 1   -1 divisors 2 2 ,(2  —  1/2). An easy computation gives At =  1/12  I  6   [-1-2 1 But At has characteristic polynomial  2(1  —  (1 + 13)/12)(A — (1 — 13)/12) and hence a diagonal Jordan form. Thus, if A and C are similar, then about the only thing that one canalways say about At and Ct is that they have the same rank. A type of inverse that behaves better with respect to similarity isdiscussed in Chapter VII. Since the generalized inverse does not have allthe properties of the inverse, it becomes important to know what propertiesit does have and which identities it does satisfy. There are, of course, anarbitrarily large number of true statements about generalized inverses. Thenext theorem lists some of the more basic properties. Theorem 1 2 1 Suppose that AEC' . Then (P1) (At)t = A (P2) (At)* _ (A *)t (P3) If leC, (IA)t = ItAt where At = - if I 0 and At = 0 if2 =0. (P4) A* = A*AAt = AtAA* (P5) (A*A)t = AtA *t    D  o  w  n   l  o  a   d  e   d   0   4   /   1   9   /   1   6   t  o   1   3   2 .   2   3   9 .   1 .   2   3   1 .   R  e   d   i  s   t  r   i   b  u   t   i  o  n  s  u   b   j  e  c   t   t  o   S   I   A   M    l   i  c  e  n  s  e  o  r  c  o  p  y  r   i  g   h   t  ;  s  e  e   h   t   t  p  :   /   /  w  w  w .  s   i  a  m .  o  r  g   /   j  o  u  r  n  a   l  s   /  o   j  s  a .  p   h  p
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