矩阵论 第五章 Hermite矩阵和正定矩阵

|yi |2
x H Ax =
i=1
|yi |2
s = 0, r = n
r
x H Ax = −
i=1
y = 0 x H Ax < 0 (4) (5) s=0 0<s<r 0 0 0 x Ax = −
H
|yi |2
s
x ∈ Cn
r i=s+1
i=1
x H Ax =
i=1
|yi |2 −
|yi |2
•
(1) (2) (3) (4) (5) •
LDU
L1 = UH 1 .
A = L1 DLH 1 dn )
L = L1 diag( d1 , · · · , L A = LLH .
•
5.2.2
A, B ∈ Cn×n ,
λ Ax = λ Bx
x ∈ Cn (3) x λ
λ . • B n
Ax = λ Bx
B−1 Ax = λx
(5.2.6)
•
Hermite
5.1
•
Hermite
Hermite
Hermite
2 3 − 2i 3 + 2i −5 −1 7 7 4 Hermite k A + pB S SH AS Hermite AB = BA Hermite
5.1.1
,
• Hermite (1) A (2) (3) (4) (5) A A A A B
f (x) = xH Ax
x = Py
Hermite (5.1.13) r−s
f (x) = xH Ax = y 1 y1 + · · · + y s ys − y s+1 ys+1 − · · · − y r yr r = rank(A), s = π (A) (5.1.13) Hermite Hermite •
H H H H PH BP = UH PH 1 BP1 U = U U = I, P AP = U P1 AP1 U = diag(λ1 , · · · , λn )
P−1 B−1 AP = diag(λ1 , · · · , λn ) B−1 A diag(λ1 , · · · , λn ), . λ1 , · · · , λn B−1 A
5.1.7 Hermite f (x) = xH Ax ) Hermite f (x) λ1 y 1 y 1 + λ 2 y 2 y 2 + · · · + λn y n y n λ1 , · · · , λn Hermite A x = Uy ( U
B
Hermite
A
•
5.1.8 f (x)
Hermite
A11 A12 AH 12 A22
, A11 ∈ Ck×k
A11 > 0
−1 A22 − AH 12 A11 A12 > 0
1 Ik −A− 11 A12 0 In−k
=
A11 0 −1 0 A22 − AH 12 A11 A12 A>0
A11 0 A11 A12 −1 AH 0 A22 − AH 12 A22 12 A11 A12 −1 A11 > 0 A22 − AH 12 A11 A12 > 0
•
λ
µ
A Ax = λx Ay = µy
x
y (1) (2)
(??) y H Ax = λy H x (??),µ = µH A = AH y H Ax = y H AH x = µy H x (λ − µ)y H x = 0 λ=µ x y
•
5.1.3
A ∈ Cn×n
A
Hermite
U
UH AU = Λ = diag(λ1 , λ2 , · · · , λn ) λ1 , λ2 , · · · , λn • 5.1.4 A ∈ Rn×n A QT AQ = diag(λ1 , λ2 , · · · , λn ) λ1 , λ2 , · · · , λn Q
5.1.1
f (x) = xH Ax Hermite x ∈ Cn x=0 x H Ax > 0 x ∈ Cn xH Ax ≥ 0 x H Ax x ∈ Cn x=0 x H Ax < 0 x ∈ Cn x H Ax ≤ 0 x H Ax x ∈ C n x H Ax Hermite f (x) = xH Ax s=r=n s=r s = 0, r = n s=0 0<s<r
1 D− 1 H 1 DD− 1 = D0 1 D− 1
)
λs ,
H
|λs+1 |, · · · ,
|λr |, 1, · · · , 1) A D0
−1 H PH 1 U AUP1 D1 = D0
•
•
(5.1.3)
D0
n
Hermite δ (A) )
A A
A ∈ Cn×n π (A), υ (A) (
•
(1) A (2) (3) A (4) (5) (6)
5.2.1 n n n n n
A
n
Hermite P PH AP PH AP = I A = QH Q S A = S2
P Q Hermite U,
(1) ⇒ (6)
A = Udiag(λ1 , λ2 , · · · , λn )UH λ1 , λ2 , · · · , λn A S = Udiag( λ1 , S n (6) ⇒ (1) Hermite n λ2 , · · · , λn )UH (5) A
A
Hermite Hermite
A
Hermite f (x) = xH Ax
A
Hermite
x1 . x= . . xn

Hermite
Hermite
•
x = Py
P
n
y1 . y= . . yn

f (x) = xH Ax = y H By B = PH AP • Hermite λ1 y 1 y 1 + λ2 y 2 y 2 + · · · + λn y n y n Hermite •
5.3
• ( B • A≥B B≥A A= A≥B • Hermite A B≥A a B b, . a ≥ b, a ≯ b, A≥B A≯B, A= A≥B, A≯B, A=B. 1 0 0 2 , ,B = a = b. A=B. 1 0 0 1 n(n ≥ 2) 4 0 0 5 , 5.3.1 B A ), A, B n A ), A>B( Hermite A≥B( B < A ). . A−B ≥ 0, A B ≤ A) ; A−B > 0, A n . B= Hermite 5 0 0 4 ” B B( ”,
x H Ax x H Ax x H Ax
5.1.9 (1) xH Ax (2) xH Ax (3) xH Ax (4) xH Ax (5) xH Ax
5.2
•
5.2.1 A
(
A n
wenku.baidu.com
)
Hermite A>0 A≥0 x ∈ Cn x ∈ Cn x=0 x H Ax > 0 xH Ax ≥ 0 A
• (1) (2) (3) (4) I>0 A>0 k>0 A>0 B>0 A≥0 B≥0 kA > 0 A+B>0 A+B≥0
5.2.7
A, B
n
Hermite
B>0,
P
PH AP = diag(λ1 , · · · , λn ), PH BP = I λ1 , · · · , λn B>0 U . P1 PH 1 BP1 = I , PH 1 AP1 Hermite
UH (PH 1 AP1 )U = diag(λ1 , · · · , λn ) P = P1 U , P
•
5.3.1 A, A1 , B, B1 , C (1) A ≥ B(A > B) (2) A ≥ B(A > B)
n
Hermite −A ≤ −B(−A < −B); n P
PH AP ≥ PH BP(PH AP > PH BP) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) A ≥ B(A > B) k k A ≥ k B(k A > k B) ; A ≥ 0, −A ≥ 0 , A = 0 ; A≥0,B≥0, A+B≥0; A ≥ B, B ≥ C , A ≥ C ; A ≥ B, A1 ≥ B1 , A + A1 ≥ B + B1 ; A ≥ 0, B > 0 , A + B > 0 . A ≥ B, B > C A>C; A > B, P n×m , PH AP > PH BP ; A ≥ B, P n×m , PH AP ≥ PH BP ; A > 0(A ≥ 0), C > 0(C ≥ 0), AC = CA , AC > 0(AC ≥ 0) .
•
5.2.6
Hermite
A A = LLH
L (5.2.3)
(??) . A n
A A
Cholesky .A A = L1 DU1 LDU
L1 , U1 di > 0(i = 1, 2, · · · , n).
D = diag(d1 , d2 , · · · , dn ) AH = A ,
H H H A = L1 DU1 = UH 1 D L1 = A
5.2.2 A n (1) Q n×m (2) |A| ≥ 0 (3) tr(A) ≥ λi (i = 1, 2, · · · , n)
•
5.2.3
n
Hermite
A ∆k > 0, k = 1, · · · , n A A
A
• •
5.2.4 5.2.5
n Hermite n Hermite
A
. A .
5.1.3 •
Hermite n x 1 , · · · , xn ∈ C
n n
f (x1 , · · · , xn ) =
i=1 j =1
aij xi xj
aij = aji
Hermite a11 a12 · · · a1n a21 a22 · · · a2n A= . . . . . . . . . an1 an2 · · · ann
5.1.2 • 1.6.8 A,B ∈ Cn×n n B = PH AP A • B A n Hermite A P
5.1.5
Is 0 0 D0 = 0 −Ir−s 0 0 0 0n−r r =rank(A) s 5.1.3 A n ( U A = UΛUH Λ A A PH 1 ΛP1 = diag(λ1 , · · · , λs , λs+1 , · · · , λr , 0, · · · , 0) = D P1 λi > 0(i = 1, · · · , s), λj < 0(j = s + 1, · · · , r) D1 = diag( λ1 , · · · ,
1 2 1 √ 8
z1 z2
• Hermite Hermite (1) (2) (3) s=r=n s=r
f (x) = xH Ax
n
s x Ax =
r i=1 H
r x=0 x ∈ Cn
n
0≤s≤r≤n y = 0 x H Ax > 0 x H Ax ≥ 0 |yi |2 x = 0 x H Ax ≤ 0 x x H Ax
A = S2 Hermite S
A = S2 = SH S
•
5.2.1
−1
A
n
λ1 , λ2 , · · · , λn QH AQ > 0
(1) A (2) Q n×m (3) |A| > 0 (4) tr(A) > λi (i = 1, 2, · · · , n) • 5.2.1 n Hermite A A= A>0 A11 Ik 0 H −1 −A12 A11 In−k A11 A12 AH 12 A22
In(A) = {π (A), υ (A), δ (A)} In(A) •A ( n Hermite ) A π (A), υ (A) A δ (A) A δ (A) = 0 (5.1.5) A B
π (A) + υ (A) = rank(A) • 5.1.6( In(A) = In(B) ) A, B n Hermite
2 f (x) = x2 1 + 6x1 x2 + x2 x1 1 1 = x2 −1 1
f (x) = xH Ax
s
y1 y2
f (x) = (y1 + y2 )2 + 6(y1 + y2 )(y2 − y1 ) + (y2 − y1 )2 2 2 2 2 2 2 = y1 + 2y1 y2 + y2 + 6y2 − 6y1 + y2 − 2y1 y2 + y1 2 2 2 2 = −4y1 + 8y2 = −z1 + z2 y1 y2 =
Hermite Hermite B Hermite Hermite AB A−1
k k
Ak p
Hermite
Hermite
Hermite
•
5.1.1 A = (ajk ) ∈ Cn×n x ∈ C n x H Ax 5.1.2 (1) A (2) A (2) A n Hermite
A
Hermite
•
(1) A (2) (3) A (4) (5) (6) •
5.2.2 n n n n
A
n
Hermite P P PH AP PH AP = A = QH Q A = S2 λ1 , λ2 , · · · , λn Q AQ ≥ 0
H
Ir 0 0 0
r = rank(A)
r Q Hermite S