-
1 13
5
N
, p- p 1 [ ]1 2x x xx = ^
px
1
pp
ipi 1
xx
=
= . (5.1)
p- :
p
0x p 0x x= = 0 px = px , . ^ p px y x y+ + p ( ). (5.1) 0 p 1
< , . Cauchy-Schwarz p-
Hlder
p qDx y x y ,
p,q 1> 1 1 1p q+ = . ,
:
1 ii 1
xx
==
1
22
i2i 1
xx
=
= ( )
iimax xx = .
, { }k 1 2x max x , x , , x = "
-
2 13
p p1
p1p
p 1p i k kp p pi 1 k k
xxlim lim x lim x 1 x xx x
x
=
= = + + + = " = ,
1k
x 1.x
<
, x ^ 1,x x 2 x ix x j , ( )i, j
1 2 12
11 1
. ^ ^ , ^
. , A ^ pAx px ^
^ , A
p
pp || || 1
p
max maxx 0 x
AxA
x == = pAx . (5.2)
(5.2) , 0x
p 0=A Ax p . , (5.2) p pAx A x p (5.3)
{ }p p pmin k : k , = \ ^A Ax x x .
5.1 pA , :
. p 0A p 0= =A A O . p pA A =
-
3 13
. p pA A+ + p V. p pAB A B p (5.4) : I. (5.2) , p 0A . p 0=A
pp|| || 1
max 0=
=x
Ax , p 0=Ax ,
. ,
p 0= =Ax Ax 0 , x ^ , =A O . II. ( )
p p pp p p p|| || 1 || || 1 || || 1
max max max= = =
= = = = x x x
A Ax Ax Ax pA .
III. ( )p p p
p p p p|| || 1 || || 1 || || 1max max maxx x x
A B A B x Ax Bx A B= = =
+ = + + = +p p .
IV. (5.3)
p p p pp pp p p
max max max
= = =x 0 x 0 x 0
ABx A Bx BxAB A A B
x x x p p
(IV) . ,
ijhH = , , . ^ iji, jmax hH = ,
1 10 1
A =
0 10 1
B =
, 0 20 1
AB = 1A B = = , 2AB = . ,
A (5.3), [ ]T1 1x = 1x = , 2Ax = .
5.2 pA (5.2) :
. p 1= . nn p pA A
-
4 13
.
p
1p
p|| || 1
1minx
AAx
== ,
p
p 1p|| || 1
1minx
AA x=
= .
: I. p
p p|| || 1max 1x
x=
= = .
II. A (5.4) 2 nn n 1 n 2
p p p pp pA A A A A A " .
III. p
p
1p 1
p|| || 1 1p p p|| || 11
p p
1 1 1max max maxmin x y 0 y 0x
A yA
Ax Ax yA yAA y
= =
= = = =
(III) 5.2
p p
1p
p p p|| || 1 || || 1
1 1min min
= == =
x x
AAx A x A p
1 .
(5.2)
pA . , p 1= p 2= p = , :
5.3 A , { }1 1 jmax =A j 1 ; (5.5) j A { }i 11 imax = A ; (5.6)i
A
( ){ }2 1 imax :A = . (5.7) : y { }i 1 k 11 imax = . x ^ ,
1 ij j ij j j iji=1 j=1 i 1 j=1 j=1 i=1
k j k1 1 1j=1
x x x
x
=
= = =
Ax
x
-
5 13
1 1k 1 1 k 11 i
1 1
maxAx Ax
A x x
= .
, [ ]Tk 0 1 0 0= k 1 k 1 1 k 1 1= = A A A ,
1 kA = 1 .
y (5.6), { }j 1 r 11 jmax = . , , x 1x = ,
1i i i i 1i i i i1 j 1 ji 1 i 1 i 1 i 1
1i i r 11 j i 1 i 1
max x , , x max x , , x
max , ,
Ax
= = = =
= =
= =
r 11maxx
A Ax
== .
[ ]T1 1 1 = , 1 = r 1 A A = . r 1A = .
y (5.7), x ^ ( ) ( ) ( )22Ax Ax Ax x A Ax A Ax x = = =D D .
A A 1 2, , , . 1 1 2 2c c cx = + + +" i iA A i= ,
( ) ( )2 2i i i i i2 2i=1 i=1
c cAx A Ax x A A x x x x x
= = = = D D D D ,
( ){ }max A A = . , 2
22
AxA
x .
2 22 2
A A A A A = = =
2
A = .
-
6 13
:
(5.5) (5.6)
1A A = .
, A ( ){ }1 i i2 1 imin : .A = A A (5.8) , (5.7) AA A A
,
( )( ){ }( ){ } ( ){ }
1 1 1i i2 1 i
1i i i i1 i1 i
max :
max : min : .
A A A
AA A A
= = =
. 3 2 4
5 2 32 1 6
A =
1 1 10 = , 2 1 5 = , 3 1 113 13 A= =
1 1 9 = , 2 1 10 = , 3 1 9 1 A 0= = .
T
38 2 392 9 839 8 61
A A =
( ) { }T 15,1322, 2.3726, 90.4952A A = 2 90.4952A = .
-
7 13
^ ^
,
ij i, j=1A
= T
11 12 1 21 2 1 = , Frobenius FA A
( )1
2, 2F ij
i, j=1trA
= = A A (5.9)
(5.9)
1 1
2 22 2
F j 2 i 2j=1 i=1
A A = = = F (5.10)
, . j i A
F 108A = .
5.4 Frobenius :
. 2 FAx A x 2 . . F FAB A B F . : I. x ^
T
2 1j j 2 j j j jj 1 j 1 j 1
2
12T 2*
11 22 ii2 i 1
1 12 2
2 * 2 22 i 2 2 i 2 F 2
i 1 i 1
x x x
.
Ax
x x x x
x x A x
= = =
=
= =
= = =
= =
"
D D " D D
. 1 2, , , k ,
[ ]
1k 2
2F 1 2 k i 2F
i 1
1 1k k2 2
2 2 2i 2 F i 2 F FF
i 1 i 1.
AB A A A A
A A A B
=
= =
= = = =
"
,
-
8 13
5.1 0 , A
p1p
1 AA
.
: p pAx A x p , Ax x=
pAx x= p . pA . 1 1A
11 1 1p p
A A .
* * *
5.2 p 1A < , n pnlim 0A = .
: nn pp 1A A < , nn ppn n0 lim lim 0A A = .
* * *
5.3 , A
FF2
A AA A+ H ,
H .
:
F FF F F
1 12 2 2 2 2 2
A A A A A H H AA A
+ = = + + H H A
F
A H= ,
(5.10), ( ) FF FH A H A H A = = .
* * *
-
9 13
5.4 A xy= , x ^ , y ^ , 1 1A x y = , 1A x y = , 2 F 2A A x y= =
2 . : (5.5)
{ }1 j i j1 11 j 1 ji 1max y x max yA x x y
=
= = = (5.6)
{ }i j i1 11 i 1 ij 1max x y max xA y
=
= = = x y . ( )22 rank 1A A yx xy x yy A A = = = ,
A A
2
2x 2
2y = , ( ) 2 22 2A A y x y y = .
, { }2 2 2ymax A AA x= = ( )F 2 2trA A A x y= = .
* * *
5.5 2A
2 22 || || || || 1
maxx y
A y Ax= =
= .
: Cauchy-Schwarz (5.3)
( ) 2 2 2 2y Ax Ax y Ax y A x y = < D 2
{ }2 2 2 2
2 2 2|| || || || 1 || || || || 1max max
x y x y 2y Ax A x y A
= = = = = .
. 0x
-
10 13
22 2|| || 1
maxx
A Ax A=
= = 0 2x , 000 2
Axy Ax=
( ) 20 20 00 0 0 2
0 2 0 2
AxAx Ax2y Ax Ax AAx Ax
= = = = .
* * *
5.6 :
. 2 2A A=
. 2 1 imaxA
= i , . A
. 22 2A A A = 2 1A = , . A
V. 22 1
A A A .
: . 5.5
( )2 2 2 2 2 2
2 2
2 || || || || 1 || || || || 1 || || || || 1
2|| || || || 1
max max max
max .
x y x y x y
x y
A y Ax y Ax y Ax
x A y A
= = = = = =
= =
= = == =
I. A A=( ) ( ) ( ){ }2 2i i:A A A A = =
(5.7) 22 i1 i 1 imax maxA i= = , . ( )i A III. 5.5,
Cauchy-Schwarz
(5.3)
( ) ( )2 2 2 2
2 2 2 2
2 || || || || 1 || || || || 1
2 22 2 2 22 2|| || || || 1 || || || || 1
max max
max max .
x y x y
x y x y
A A y A Ax Ax Ay
Ax Ay A x y A
= = = =
= = = =
= =
D
=
0x 2A Ax= 0 2 , 0y x=
2 20 0 0 22
x A Ax Ax A = = ,
-
11 13
0x2
2 0 0 2A A x A Ax A = = .
, , A 2 22 1 1A A= = . V. (5.7) 5.1
( )2 max2 11 1A A A A A A A A = = 1A .
* * *
5.7 2U AV A = 2 , UU I = . V V I = :
2 2
22U Ax x A UU Ax x A Ax Ax = = =
2 2
2 2 2 2|| || 1 || || 1max maxx x
A Ax U Ax U = =
= = = A .
, 2 22 2
Vx x V Vx x x x = = =
( )2 2
2 2 2|| || 1 || || 1max maxx x
AV AVx A Vx A= =
= = = 2
22 2U AV U A A = = .
* * *
5.8 :
. ( )1 2diag ,A A A= , { }2 1 2 2max ,A A A= 2 , . ,
iiO H
BH O
= 2 2B H= .
: . ( )1 1 2 2diag ,A A A A A A = ,
-
12 13
( ) ( ){ }( ) ( ){ } { }
2 i i 1 1 2 2i
i i 1 1 i 2 2 1 2 2 2i
max :
max : max , .
A A A A A
A A A A A A
= = =
. (diag ,B B HH H H = ) , HH H H
( ){ }2 i iimax :B H H= = 2H .
* * *
A ( ) 1p p=cond A A A 5.1
( ) 1cond A . , (5.7) (5.8) p 2=
( ) maxmin
= A A
A A
cond A .
=Ax , x ( )+ = + A x x
( ) = A x .
,
pp
xx
, A
.
-
13 13
,
( )
( )
1 1 1p p pp p p p
p p p p
p
p
=
A A A A x
x x x
cond A
(5.11)
p p p= Ax A x p 1p p px A .
1
pp p p p
p p p
A A x A x
xp (5.12)
1 1p pp p
= x A A 11p p p
x A .
, (5.11) (5.12)
( ) ( )p p
p p
1 pp
x cond A
cond A x .
. ( ) 100=cond A , 2 1= , p 0,1 =
2
2
10,001 0,1 100 0,1 10100
= =xx
.
