矩阵分析与应用课件_张贤达_第10讲(2010)

W Hvec(Ak )
2 2
=
i=1
=
i=1
2 |uH A u | k i i
K
n 2 |uH i A k ui | i=1 2 F
28 64
ˆ = arg max U U k=1
Hermitian
A, B U ΛA , ΛB Λ
[37],[446]
1 Λ = ΛA Λ− B
τ1, · · · , τK ∀1 k k p p
24 64
K
V
m×n
V HRx(τk )V = diag[d1(k ), · · · , dn(k )], di(k ) = dj (k ), ∀1 i=j n,
A
20
64
1 2 n
A E s(t)sH(t − τ ) = D τ
(m ≥ n )
3 Rv (τ ) = E{v (t)v H(t − τ )} Rv , τ = 0 ( = δ (τ )R v = O, τ = 0 (
) )
21 64
4 s(t) E s(t)v H(t − τ ) = O
v (t)
ak = 0
(ak , aj ) = 0, (k = j )
Hadamard’s inequality 2
n n
A = (ai,j ) ∈ Cn×n
11 64
|λi| = |detA| ≤
i=1 i=1
ai,i
ai,j = 0, (i = j )
8.6
Jordan
A ∈ Cn×n
1 A
A A ∈ Cn×n AHA = AAH 1 A = U ΛU H , U U HAU = Λ
k =1
wk Ak − U Λk U H
2 F
w1, · · · , wK
wk = 1 U
25
64
Λ1 , Λ2 , · · · , ΛK U
U HU = I 1
[472] K
J (U , Λ1, · · · , ΛK ) = min
U ,Λk
Ak − U Λk U H
k =1
2
2 F
H n ×n W = [vec(u1uH 1 ), · · · , vec(un un )] ∈ C vec(ABC ) = (C H ⊗ A)vec(B ) W = [u∗ 1 ⊗ u1 , · · · , u∗ (A ⊗ B )(C ⊗ D ) = (AC ) ⊗ n ⊗ un ] (BD ) W HW = I n 2
A
m∞ i,j
λ = n · max |ai,j |
m∞ m∞
1 A + AH 2 1 |Im(λ)| ≤ A − AH 2 |Re(λ)| ≤ x A Ax = λx λ xH
λ = xHAx
9 64
|λ| = |xHAx| ≤ A m∞ 1 1 1 |Re(λ)| = |λ + λ∗| = |xH(A + AH)x| ≤ A + AH 2 2 2 1 1 H 1 H ∗ |Im(λ)| = |λ − λ | = |x (A − A )x| ≤ A − AH 2 2 2
A
3
64
k →∞
A 1 1 1 ≥ ≥ ··· ≥ λn λn−1 λ1 A A−1 v = v0 = 0 0 v k = A−1v k−1 vk vk = max(v k ) un max(un) (Av k = v k−1)
4
64
k →∞
lim v k =
k →∞
lim max(v k ) =
22 64
Rx(0) Rx(τi), τi = 0
A τi = 0 τ1 = 1, · · · , τK = K
A R (τ i ) A
K A1 , · · · , AK K A k = U Λk U H , U k = 1, · · · , K joint diagonalizer U
23 64
(λ I − A ) →
d1(λ)|d2(λ), · · · , ds−1(λ)|ds(λ), s ≤ n 1, · · · , s) di(λ) (λ I − A ) di(λ) (λI − A)
di(λ) (i = 1
2 (λI − A )
di(λ)
16 64
3 J i (λ i ) = Jordan λi 1 λi
(i = 1, 2, · · · , s) 1 λi mi×mi
J = diag[J 1(λ1), J 2(λ2), · · · , J s(λs)].
P P = [P 1 P 2 · · · P s], P i = [pi1 pi2 · · · pimi ]n×mi
b1n b2n =B . . bnn
⇒ B HB = U HAHU U HAU = U HAHAU = U HAAHU = U HAU U HAHU = BB H B
13 64
b12 = 0,
b13 = 0, · · · , b23 = 0, · · · , ...
b1n = 0 b2n = 0 . . bn−1,n = 0
t = 1, 2, · · ·
19
64
W y (t) = W x(t) = W As(t) + W v (tБайду номын сангаас s(t)
WA = I A
n
W = A†
n
x(t) =
i=1
aisi(t) + v (t) =
j =1
aj [αj sj (t)] + v (t) αj
A W WA = I WA = PD = G s(t)
Rx(τi) = E x(t)xH(t − τi) = E [As(t) + v (t)][As(t − τi) + v (t − τi)]H = AE s(t)sH(t − τi) AH + E v (t)v H(t − τi) = AD 0AH + Rv , AD τi AH, τi = 0 τi = 0
(λ − λ 1 ) m 1 , ( λ − λ 2 ) m 2 , · · · , ( λ − λ s ) m s λ1, λ2, · · · , λs m 1 , m2 , · · · , ms m1 + m2 + · · · + ms = n (λ − λ i )m i ... ... Jordan
K
I − W (W HW )−1W H vec(Ak )
2 2
2 2
= arg max W (W HW )−1W Hvec(Ak ) U k=1
K
= arg max W W Hvec(Ak ) U k=1
K
2 2
27
64
= arg max W Hvec(Ak ) U k=1
2 2
n
n H ( u∗ ⊗ u ) vec(Ak ) i i 2 2
P −1AP = J ⇒ AP = P J ⇒ AP i = P iJ i(λi) (A − λiI )pi1 = 0 (A − λ I )p = p i i2 i1 ⇒ . . (A − λiI )pi,mi = pi,mi−1 ⇒ (A − λiI )j pij = 0, (j = 1, 2, · · · , mi) pi1, pi2, · · · , pi,mi Pi P
2010
11 26
·
6A314
1
64
Email: zxd-dau@tsinghua.edu.cn : FIT 3-117 : 010-62794875
8
(3)
8.5 8.6 8.7
2 64
Jordan
8.8
8.5
A.
A
v0 v = v0 = 0 0 v k = Av k−1 vk vk = max(v k ) u1 lim v k = k →∞ max(u1) lim max(v k ) = λ1
n n
A = (ai,j ) ∈ Cn×n
|λi|2 ≤
i=1 i,j =1
|ai,j |2 = A A
2 F
AHA = AAH,
,
Hadamard’s inequality 1 λi
n n
A = (ai,j ) ∈ Cn×n
n
|λi| = |detA| ≤
i=1 j =1 i=1
|ai,j |2 ,
... ...
1 λi mi×mi
14
64
A
λi (i = 1, · · · , s) Jordan λ i λ j ( j = i) λi Jordan
mi
Jordan 1
J (λ I − A ) d1(λ) d2(λ) ... ds(λ) 0 ... 0
15 64

λ Gerschgorin
A 1 i |ai,j |
j =i
7 64
|λ − ai,i| ≤ Ri = λ = 0 detA = 0
|ai,i| ≤ j =i |ai,j | λ = 0
A λ
A ∈ Cn×n, x ∈ Cn |xHAx| ≤ A |xHAx| |xHAx| =
i,j m∞
x
2
=1
1 λn
♣
/ Onion-Peeling
B.
Gerschgorin 1 λi
n
A = (ai,j )n×n
|λ − ai,i| ≤
j =1,j =i
|ai,j |
(i = 1, 2, · · · , n)
Gerschgorin 2 k
A k
5
64
1 2
3
6 64
4
n A = (ai,j )n×n R i (A ) = j =1,j =i |ai,j |, i = 1, 2, · · · , n |ai,i| > Ri, i = 1, 2, · · · , n |ai,i| ≥ Ri, i = A 1, 2, · · · , n i0 |ai0,i0 | > Ri0 A
m∞ m∞
Hermitian
Hermitian Cn×n ··· ... ... ∗ . . ∗ λn
10 64
Schur’s Lemma λ1, λ2, · · · , λn U −1AU = U HAU =
A ∈ U λ1 ∗ λ2

Schur’s inequality λi
26 64
min Ak − U Λk U H
2 F
= min Ak −
i=1
dk (i)uiuH i
F 2 2
= min vec(Ak ) − W dk dk = [dk (1), · · · , dk (n)]T
dk = (W HW )−1W Hvec(Ak )
K
ˆ = arg min U U k=1
A H = U Λ∗ U H
12
64
⇒ A H A = U Λ∗ U H U ΛU H = U Λ∗ ΛU H = U ΛΛ∗U H = U ΛU HU Λ∗U H = AAH A
2
A U U HAU =
AHA = AAH b11 b12 b22 ··· ··· ...
Schur
∃ k, 1
di(k ), i = 1, · · · , n A m×n M . M = N , M = N G,
R x (τ k ) A† N G n×n
V
B.
N ×N Λ1, · · · , ΛK
K
A = {A1, · · · , AK } U ∈ CN ×N K
J (U , Λ1, · · · , ΛK ) =
17
64
1.
−1 1 0 A = −4 3 0 Jordan 1 0 2


2.
Jordan dx 1 dt dx2 dt dx3 dt
= − x1 + x2
18 64
= −4x1 + 3x2 = x1 + 2x3
8.7
A.
x(t) = As(t) + v (t), x(t) = [x1(t), · · · , xm(t)]T s(t) = [s1(t), · · · , sn(t)]T v (t) = [v1(t), · · · , vm(t)]T A ∈ Rm×n (m ≥ n)
i,j
= n · max |ai,j |
ai,j x∗ i xj ≤ max |ai,j | ·
i,j i,j
|xi||xj |
8 64
≤ max |ai,j | ·
i,j
1 2
(|xi|2 + |xj |2)
i,j m∞
1 = max |ai,j | · (n + n) = A i,j 2
A ∈ Cn×n |λ| ≤ A
B = diag(b11, b22, · · · , bnn)
2 A ∈ Cn×n A 3 P −1AP = J = J i (λ i ) = λi 1 λi A ∈ Cn×n Jordan J 1 (λ1 ) J 2 (λ 2 ) ...
A
n
P J s (λ s )