Page 460 - Advanced Linear Algebra

444 Advanced Linear Algebra

im(W ) im(W)
*

u v
1
W(u )=s v 1
k k
k
ONB of u W (v )=s u v ONB of
eigenvectors r k k k r eigenvectors
for W*W for WW*
u r+1 W v r+1

u n W v m

ker(W) ker(W )
*
Figure 17.1

For ~ Á Ã Á , the positive numbers ~ j are called the singular values

of . If we set ~ for € , then

i
"~ "

for ~ Á Ã Á . We can achieve some “symmetry” here between and i by

setting # ~ ² ° ³ " for each  , giving
# 

"~ F

€
and
" 
i
#~ F

€
The vectors #Á à Á # are orthonormal, since if Á  , then

º# Á # »~ º " Á " »~ º i " Á " »~ º" Á " »~ Á

iž

Hence, ²# ÁÃÁ# ³ is an orthonormal basis for im ² ³ ~ ker ² ³ , which can be

extended to an orthonormal basis 9 ~²# Á Ã Á # ³ for = , the extension

ÁÃÁ# ³ being an orthonormal basis for ker ² ³. Moreover, since
i

²# b

i
#~ " ~ #

the vectors #Á Ã Á # are eigenvectors for i with the same eigenvalues
~ as for i . This completes the picture in Figure 17.1.