332 CHAPTER 13. MATRICES AND THE INNER PRODUCT
60. Show that A+ = (A∗A)+ A∗. Hint: You might use the description of A+ in terms ofthe singular value decomposition.
61. Let A =
(1 −3 03 −1 0
). Then
−√
2/2√
2/2 0√2/2
√2/2 0
0 0 1
T
AT A
−√
2/2√
2/2 0√2/2
√2/2 0
0 0 1
=
16 0 00 4 00 0 0
AAT =
(10 66 10
). A matrix U with
UT AATU =
(16 00 4
)
is
( √2/2 −
√2/2√
2/2√
2/2
). However,
( √2/2 −
√2/2√
2/2√
2/2
)T (1 −3 03 −1 0
) −√
2/2√
2/2 0√2/2
√2/2 0
0 0 1
=
(−4 0 00 2 0
).
How can this be fixed so that you get
(4 0 00 2 0
)?
332 CHAPTER 13. MATRICES AND THE INNER PRODUCT60. Show that At = (A*A)* A*. Hint: You might use the description of At in terms ofthe singular value decomposition.61. LetA= tr -3 0 . Then3 -1 0_ 2/2 v2/2.0\" — 2/2. v2/2 0V2/2 V2/2 0 | ATA] V2/2 V2/2 00 0 1 0 0 116 0 0= 0 4 00 0 01war =( 0 6 ) ama 0 wa6 101UTAA'U = 6 00 4i ( v2/2 —v2/2 ) Howeris V3/2 3/2T —VJ2/2 2/2 0V2/2 —V2/2 1 -3 0(93; V2/2 (; -1 | v2/2 v2/2 00 0 1_ 40 07 0 20/)°4 0 0How can this be fixed so that you get ( 020 ) ?