'+ . /. . //.
/. ..
9/ ;
#
#$ #)
-)- ) -#) )- ) -#)$ # -- $#
8# -) -#)
-
l
) -
-
l × l
l, m N
m )
-#
#)@
Ml,m (R)
l = m
) ) -#) < -) #$) Ml (R) ) ) -#) -# 8# #) 8# -- -# · -# Ol,m l = m Ol $)# -# - Il -# --$ · · (ai )16i6l A, B Ml,m (R) A Ml,m (R) - -) ) -) )- ) #$) -# (a1 , . . . , al ) -# -# a 1 , a2 , . . . , a l hA, BiF = -#(AT B) R - D -#(M ) $) -# kAkF = p hA, AiF h , iF Ml,m (R) )- ) -# #$ ) # 8# -- -# 8# -- k kF A Ml,m (R) - A $- #- )# )# Ml,m (R) -# #(A) # -# # @ #(AT ) k (R) Ml,m A - @ lm )-<# = #(A) ) ) -#) A Ml,m (R) -) @ = k p N -# k N #)@ Ml (R) -)# )) $)-#- @ #(A) 8# -#))$ # -- -# ) · -# - ) -) )- 8# -) M · -# 8# -- Ml (R) A Ml,m (R) AT · - kxk2 Rp ) )-#-# @ - # -- -# # x x Rp !"! % n, p, q N %&' %&' '%&%% '%' % A, B Mn,p (R) % C Mp,q (R) & &'' hA, BiF % ' %' A % B % u Rp %&& 6 kAuk2 6 kAkF kuk2 . %&& 6 kACkF 6 kAkF kCkF !)! !+ k (R) '% % ':& %&' %&' n, p % k '%&%% '%' %' 6 Mn,p k (R) A %& Mn,p k (R) %&& 6 k 6 min(n, p) % 6 & %% R A Mn,p % S = AAT % S = AT A C&& 6 S '% %& 'C%&6 6 % 6 ' &' &&' '%' ' %&& 6 (A) = (S) % u Rn %& && S & & && > 0 % '% v =
AT u/ Rp %&& 6 v '% %& && S & & &&
% kvk2 = kuk2
%&& 6 '% U Mn,k (R) % = (1 , . . . , k ) Mk (R) %' 6
S = U U T 1 > . . . > k > 0 % U T U = Ik
%&& 6 (S) = (U ) % 6 U U T '% %& &% &%
'& (U ) ' Rn
'% V = AT U D Mp,k (R) M D = (1/ 1 , . . . , 1/ k ) Mk (R)
%&& 6 V T V = Ik % S = V V T
C& 6
A = U V T ,
= ( 1 , . . . , k )
% )*
# && (& #&-(
&( ( k ( &( ( - && (& #& -& # (&#
#&#
k (R) &( ( k : n, p & k #& # &(# #&(&& #&#
& A Mn,p
k 6 min(n, p) #=( -#& A = U V T #&(& # (=( (&
& l N & Ve Mp,l (R) &# > l < k & Ve T Ve = Il & (v1 , . . . , vl ) (Rp )l # # Ve & (v1 , . . . , vk ) (Rp )k # # V -(( > kA - AVe Ve T k2F = kAk2F - kAVe Ve T k2F
&(( >
: h ,
i2
l
X
hvh , vm i22
1 6 j 6 l
#
k
X
kAVe Ve T k2F =
h=1
h
m=1
!
-# (& #( # #( Rp
F( &&
l+1 6 i 6 k
Pl
bj = 1 - m=1 hvj , vm i22
& &&
ai =
Pl
2
m=1 hvi , vm i2
&
&(( > # (ai ) & (bj ) #& # (-# #&# & > Pki=l+1 ai 6 Plj=1 bj
&(( > kAVe Ve T k2F 6 Plh=1 h & > -&- # & #& #
&({v1 , . . . , vl }) = (Ve ) : &(X) -#
Pk
l (R) &(( > kM - Ak2 >
& M Mn,p
h=l+1 h -&- # & #
F
T
& # M = U V : = ( 1 , . . . , l ) U (# V #& &(
(- # l (=(# # U (# V
!$% !*
% p, k %)* *%)%% *%* % V Mp,k (R) % 0 V T V = Ik 2) %%
W Mp,k (R) % MV,W %) Mp+k (R) 5 ) * )
MV,W =
V
Ip
Ok W T
.
** 0 W T V *% %) )*
-1
%)) 0 MV,W *% )* %) * )* MV,W
%)) 0 )% (W ) (W ) % (V ) *% ****
*5%)* * Rp (W ) (V ) = Rp
& ,, , , 0,, 2 , z Rp 4 z (W )
,4 W T z = 0
5% %)
PV,W = (V
-1
Op )MV,W
Ip
Ok,p
.
%)) 0 PV,W *% %) )% *) (V ) )E% F (W )
% q N %)) 0 * * %)* )** Mq (R) *% )% %
0 % M 7 M -1 *% % *) % )%
%)) 0 *% * V V * Mp,k (R) % 0 W T V *% )* )
%% W V % % W 7 PV,W *% % V * Mp (R)
*!% !*
% n, p % k %)* %)* *%)%% *%* %* 0 k 6 min(n, p) 5% ) %%
*% * %)
E = Mn,k (R) × Mk (R) × Mp,k (R) .
k (R) %) ) k % (U, , V ) E %* 0
% A Mn,p
A = U V T , U T U = V T V = Ik
% F %* *%)%% *%* *% (U, , V ) 5%5
%)5 * )E) )%
( (U , , V ) E +-. . : R Mn,p (R) 3 . (t) =
(U + tU )( + t)(V + tV )T
(.. 9 + (+ t 7 .(U + tU ) t 7 .( + t) ( t 7 .(V + tV ) +(
+((+ + t = 0
k (R) + t = 0
3. 9 (t) Mn,p
(.. 9 +( 3( 3. +. R ( . .++ 3.3
(0) 0
( TA = {U V T + U V T + U V
T
T
T
| (U , , V ) E , U U = V V = Ok }
k (R) A (
3.. 9 (+ + 33(+ TA +( + (.+ ((+ D Mn,p
9 TA +( +++ (. Mn,p (R) ( . +
T
( NA = {N Mn,p (R) | N U = Op,k , N V = On,k } (.. 9 NA +(
+++ .( D TA + Mn,p (R) . .( +. h , iF
( A Mn,p (R) ( 9 A 3. ( +
(C) (AVVT ) = (A) ( (AT UUT ) = (AT ) .
(.. 9 + A 3. ( .+ .(A) 6 k (
(AT U U T ) = ker(A) .
k (R) (. A 3.
(.. 9 +( > 0 ( 9 . (( A Mn,p
( -+ 9 kA - AkF 6
( : Mn,p (R) Mn,p (R) × Mp,n (R) 3 . (A) = (AV V T , AT U U T ) . ((
A Mn,p (R)
(. ker() ( NA (.( D 9+(
( A : Mn,p (R) Mn,p (R) .( .( +. TA + Mn,p (R)
(.. 9 = A
k (R) 3.( ( ( W = AT U U T (.. 9
( A Mn,p
+ PV,W +( (. .( +. (V ) .-( D (W ) .+
A = AV V T PV,W .
k (R)B(A, ) +( (
3. 9 +( > 0 ( 9 .+(.( A D Mn,p
M B(A, ) = {A Mn,p (R) | kA - AkF < } +( .( Mn,p (R) (.3 A ( . & A *& *& 1* NA 1 Mn,p (R) &** * && A Mn,p (R) A (A) = (In - U U T )A(Ip - V V T ) &** 9 A (AB) = 0 * && B Mp (R) k (R) ;*& & & A Mn,p &** 9 1 W = AT U U T A (A) = (In - U U T )(A - A)V V T (PV,W - PV,V )(Ip - V V T ) . ;* 9 kA (A)kF 6 p (n - k)k(p - k) kA - AkF kPV,W - PV,V kF k (R) A &** 9 TA 1& && 1 1 &*1 &&1 C Mn,p
