Kenneth Kuttler

378 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Myx

A+(y)

ker(A∗A)

Then A+ (y) is as given in the picture.

Proposition 13.18.2 A+y is the solution to the problem of minimizing |Ax−y| for all xwhich has smallest norm. Thus∣∣AA+y−y

∣∣≤ |Ax−y| for all x

and if x1 satisfies |Ax1−y| ≤ |Ax−y| for all x, then |A+y| ≤ |x1| .

Proof: Consider x satisfying 13.33, equivalently A∗A x= A∗y,(σ2 00 0

)V ∗ x=

(σ 00 0

)U∗y

which has smallest norm. This is equivalent to making |V ∗x| as small as possible becauseV ∗ is unitary and so it preserves norms. For z a vector, denote by (z)k the vector in Fk

which consists of the first k entries of z. Then if x is a solution to 13.33(σ2 (V ∗x)k

(σ (U∗y)k

)

and so (V ∗x)k = σ−1 (U∗y)k . Thus the first k entries of V ∗x are determined. In order tomake |V ∗x| as small as possible, the remaining n− k entries should equal zero. Therefore,

V ∗ x=

((V ∗x)k

(σ−1 (U∗y)k

(σ−1 0

0 0

)U∗y

and so

x=V

(σ−1 0

0 0

)U∗y ≡ A+y ■

Lemma 13.18.3 The matrix A+ satisfies the following conditions.

AA+A = A, A+AA+ = A+, A+A and AA+ are Hermitian. (13.35)

378 CHAPTER 13. MATRICES AND THE INNER PRODUCTker(A*A)Then A* (y) is as given in the picture.Proposition 13.18.2 A*y is the solution to the problem of minimizing |Ax — y| for all xwhich has smallest norm. Thus|AA*y—y| < |Axw — y| for all xand if x, satisfies |Ax, —y| < |Ax —y| for all x, then |ATy| < |a|.Proof: Consider x satisfying 13.33, equivalently A*A x = A*y,2o 0 Vig o 0 U*y0 O 0 0which has smallest norm. This is equivalent to making |V*a| as small as possible becauseV* is unitary and so it preserves norms. For z a vector, denote by (z), the vector in Féwhich consists of the first k entries of z. Then if x is a solution to 13.33[oer y= ( 2m )0 0and so (V*x), =o! (U*y),. Thus the first k entries of V*x are determined. In order tomake |V*a| as small as possible, the remaining n — k entries should equal zero. Therefore,ven (MP) (M8 8 uy0 0 0 Ooav(* U*y=Aty Eland so0 OLemma 13.18.3 The matrix A‘ satisfies the following conditions.AAtA =A, AtAAt = A*, ATA and AA® are Hermitian. (13.35)