期末復(fù)習(xí)-矩陣分析課件

簡單性質(zhì):k(AB)

(kA)B

A(kB);A(BC)

ABAC,

(BC)A

BACA;(AB)(CD)

AC

AD

BC

BD;A(BC)

(AB)C

ABC;EmEn

EnEm

Emn;若A,B均為上(下)三角陣,則AB也為上(下)三角陣;(AB)H

AHBH;若A,B均為Hermite矩陣,則AB也為Hermite矩陣;若A,B均為方陣,則trace(AB)

trace(A)trace(B)定理:設(shè)A

(aij)mn,B

(bij)pq,C

(aij)ns,D

(bij)qt,則:(AB)(CD)

(AC)(BD)c

D

a

B

c

D

ansmnn1m1a1n

B

c11Dc1s

D

M

M M

L

a11B證明:(AB)(CD)

L.M

M.B

Lnna1k

cks

BD

namk

cks

BD

k

1nk

1M

(

AC)

(BD)amk

ck1BD

k

1

k

1

a1k

ck1BDMLML定理:若A,B均為正規(guī)矩陣,則AB也為正規(guī)矩陣證明:設(shè)AAH

AHA,BBH

BHB,則:(AB)(AB)H

(AAH)(BBH)

(AHA)(BHB)

(AHBH)(AB)

(AB)H(AB)定理:若A為m階可逆方陣,B為n階可逆方陣,則AB也為可逆方陣,且(AB)1

A1B1證明:(AB)(A1B1)

(AA1)(BB1)

EmEn

Emn定理:rank(AB)

(rankA)(rankB)證明:設(shè)rankA

r,rankB

s,則存在非奇異矩陣M,N,P,Q,使得:

0

0

0

011PBQ

B

EsMAN

A

Er0,

0從而A

M1A1N1,

B

P1B1Q1,(AB)

(M1A1N1)(P1B1Q1)

(M1P1)(A1B1)(N1Q1)

(MP)1(A1B1)(N

Q)1

rank(AB)

rank(A1B1)

rs

(rankA)(rankB)定理:{x1,x2

,L,xn}及{y1,y

2

,L

,yq}均線性無關(guān)的充分必要條件是{xiyj,

i

1,

2,

L

,

n;

j

1,

2,

L

,

q}線性無關(guān).證明:令A(yù)

(x1,x

2

,L

,xn)Cmn,B

(y1,y

2

,L

,yq)Cpn,C

(x1y1,

L

,

x1yq,

x2y1,

L

,

x2yq,

L

,

xny1,

L

,

xnyq)Cmpnq,則AB

C.由于rank(C)

rank(AB)

rank(A)rank(B),結(jié)論顯然.定理:設(shè)A為m階方陣,B為p階方陣,則|AB|

|A|p|B|m.證明:

設(shè)A,

B的Jordan

為JA和JB,

則存在非奇異矩陣P和Q,

使得

A

PJAP1,

B

QJBQ1.

從而:AB

(PJAP1)(QJBQ1)

(PQ)(JAJB)(P1Q1)

(PQ)(JAJB)(P

Q)1

|AB|

|JAJB|

ppp

pp

A

p

B

mmj

11

2

m

j

j

1j

1

j

1p

p

1

j

2

j

L

L

m

j定理:設(shè)A為m階方陣,B為p階方陣,則存在一mp階置換矩陣P(每一行和每一列都恰好有一個1,其余元素為0,滿足

PTP

PPT

Emn),使得:PT(AB)P

BA證明:容易驗證,存在mp階置換矩陣,使PT(AEp)P

EpAPT(EmB)P

BEm從而:PT(AB)P

PT(AEp)(EmB)P

PT(AEp)PPT(EmB)P

(EpA)(BEm)

BA定理:設(shè)A為m階方陣,B為p階方陣,則:AB

~

BA定理:設(shè)A

(aij)mn,B

(bij)np,則:(AB)[k]

A[k]B[k]其中:A[k]

AAALAk個A為A的關(guān)于Kronecker積的冪.第二節(jié)

函數(shù)矩陣對矩陣的導(dǎo)數(shù)定義：設(shè)A(aij)mn,B(bkl)pq,若A是B的函數(shù),且aij對所有bkl可以求偏導(dǎo)數(shù),則稱函數(shù)矩陣A對矩陣B可求導(dǎo)數(shù),

A

AbA

b

b1q

11

12Lklkl

mnb

aij

bApq

mpnqA

b

b

bbA

A

A

b

b,其中DBDAp1

p

22q

2221M

MLM

A

A

ML稱DA

為函數(shù)矩陣A對矩陣B的導(dǎo)數(shù).DB性質(zhì)1：設(shè)A(aij)mn,B(bij)nr,C(cij)pq,則:qpD(

AB)

DA

(EDC

DC

A)

DBDC

B)

(E推論1：若A為常數(shù)矩陣,則:

A)DB

,DC

DCD(

AB)

(Ep若B為常數(shù)矩陣,則:qD(

AB)

DA

(EDC

DC

B)推論2：若A,B都是一階矩陣,則:pn

E

)

DB

k

DB

D(kB)

D((kE)B)

k(EDC

DCDC

DCD(AB)

DA

B

A

DBDC

DC

DC推論

3：若A(a1,

a2,

L

,

am)T,

B(b1,

b2,

L

,

bm)T,

C(c1,

c2,L

,

cp)T,則:DC

DC

DC

DCT

DCT

DCTD(

AT

B)

BT

DA

AT

DB D(

AT

B)

DAT

DBTB

A,性質(zhì)2：設(shè)A(aij)mn,B(bij)pq,C(cij)rs,則:DB

D(

A

B)

DA其中:

B

A

Dc

DC

DC

rsij

A

DcA

DB

DcA

DBDc

A

DBDcA

DB

DcA

DBDc

A

DBA

DB

A

DB

A

DBDcDB

r

2r12

s

2221Dc1sDc12Dc11MMMLMLij

L證明：D(

A

B)

D(

A

B)

D(

A

B)

D(

A

B)D(

A

B)rsDcDcr1Dc1sM

Dc11

MDCLMLD(a1n

B)

D(a11B)D(

A

B)DcijDcij其中:

ij

ijijijijijijijijij

B

aB

amnDcDB

DB

DBDc

DcDc

DcDcD(amn

B)

DcDcij

m1

Dc1nDc11

Dc

Dam1

BDa1n

Da11

B

D(am1

B)MML

aM

DBL

aM

DamnMLMLMLMLMB

D(

A

B)

D(

A

B)

D(

A

B)

D(

A

B)D(

A

B)DCDc1sDa1nB

Dc1s

Da11Da1nrsDc11

Dc11

Da11B

L

B

LM

M

M

L

M

M

MDcDcr1Dc1sM

Dc11

MLML

rsrs

A

B

B

DcijDB

Dc

DcDa

Dcrs

Da

Dc

DcDa

Da

DcDamn

DcDcDamn

B

Dc

Dam1

B11

Br1r1

Dam1

B11

B

Dcr11s1s

Dam1

B1111

Dam1

BB

1nDcrsMDamn

B

1nDcr1M

L

MDamn

B

MMLMLMMLMLLLij

DcDB

B

A

DCDA性質(zhì)3：設(shè)函數(shù)矩陣A對矩陣B可導(dǎo),則:,,

DAH

DA

H

DB

DBH

DAT

DAT

DB

DBT例

1：純量函數(shù)f(X)對向量變量X

(x1,

x2,

L

,

xn)T的導(dǎo)數(shù).

gradf

(x)DXDf

x1

x2

f

f1

xn

f

T

n

x

f

f

x

f

x

2

M

L例

2：f(x)

TX,

其中X

(x1,

x2,

L

,

xn)T為向量變量,

(a1,a2,L

,

an)T常數(shù)向量,則:

TT1

2

n

a

,

a

,

L

,

a

n

xx

x

DXDf

Df

TDf

f

f

f

T2

1LDX

T

DX

例

3：f(x)

XTAX,

其中X

(x1,

x2,

L

,

xn)T為向量變量,

A

(aij)為n階常數(shù)矩陣,則:f

T

xn

L

DX

x1

x2Df

f

fnjjjxa x

xkl

k l

aklx

xf

nk

,l

1k

,l

1(xk

xl

)其中:nx

akl

k

,l

1

xk

(xl

)nj

n

n

akj

xk

ajl

xlk

l

1n

kj

xlk

,l

1

akl

lj

xkx

(akj

a

jk

)xkk

1(xk

)xl

j

2

AX

(

AT

A)X

,xn

f

TDX

x1

x2Df

f

f

LA對稱時TTTTDXT

TD((A

Y

)

X

)DXDXT

TD((A

Y

)

X

)DXD(

X AY

)DXD(

X AX

)

(

AT

A)X

AY

A

Y

D((AY

)

X

)Y

XY

XY

X例

4：

X(t)

(x1(t),

x2(t),

L

,

xn(t))T,

f(t)

f(X(t)),

則:

Df

T

dX

Df

dX

n

xn

dtdt

x1

dt

x2

dt

f

f

f

dXdf(t)

f

dx1

f

dx2

dt

DX

dt

DX

dtdxn

L

f例5：f(X)

trace(XXT),其中X

(xij)mn,則:xijm

nkl

ki

ljm

nijklm

nijklm

nijm

ndxdxDXDDX

DXDf

D traceXXT

dx2x2

2x

2X

kl

dx

2

dx

dmnmn

k

1

l

1mn

k

1

l

1mn

k

1

l

1mnk

1

l

1

kl

2x

klk

1

l

1

2x

第三節(jié)

Kronecker積的特征值考慮如下復(fù)系數(shù)二元多項式:設(shè)A為m階矩陣,B為n階矩陣.定義:f

(x,

y)

li

j

ijc x

yi,

j

0ljic

A

Bf

(

A,

B)

iji,

j

0定理：設(shè)ACmm,B,x1,x2,L

,

xm是A的屬于特征值1,2,

L

,

m的特征向量;

y1,y2,

L

,

yn是B的屬于特征值1,

2,L

,

n的特征向量.

則:

f

(r

,

s

), (r

1,

2,

L

,

m;

s

1,

2,

L

,

n)是

f(A,B)的特征值，xr

ys

, (r

1,

2,

L

,

m;

s

1,

2,

L

,

n)

是相應(yīng)的特征向量。證明：Axr

rxr,

r

1,

2,L

,

m;Bys

sys,

s

1,

2,L

,

n.r

r

s

s

sAi

xr

i

x

;

Bi

y

i

y

.i

jij

r

s

r

ssjiijrc

A

x

B

y

jiijr

si

jijx

y

c

A

B

x

y

y

c

A

B

r

s

rf

(

A,

B)xc

s

rx

y f

(

,

s

)xr

y

li,

j

0s

li,

j

0li,

j

0li,

j

0推論1：rs

(r1,2,L,m;s1,2,L,n)為AB的特征值，xrys(r1,2,L,m;s1,2,L,n)為相應(yīng)的特征向量.推論2：rs

(r1,2,L,m;s1,2,

L

,n

)為AEnEmB的特征值,xrys

(r1,2,L,m;s1,2,L,n

)為相應(yīng)的特征向量.AEnEmB—A與B的Kronecker和第四節(jié)

矩陣的列展開與行展開定義：設(shè)A

(aij)mn,

則稱

rs(A)

(a11,

L

,

a1n,

a21,

L

,

a2n,L

,

am1,

L

,

amn)為A的行展開,

稱

cs(A)

(a11,

L

,

am1,

a12,

L

,am2,L

,

a1n,L

,

amn)T為A的列展開.定理：設(shè)A

(aij)mn,

B

(bij)np,

C

(

q,

則:rs(

ABC)

rs(B)(

AT

C);

cs(

ABC)

(CT

A)cs(B).推論1：設(shè)A

(aij)mn,B

(bij)np,則:rs(

AB)

rs(Em

AB)

rs(

A)(Em

B);T

rs(

ABEp

)

rs(B)(

A

Ep

).推論2：設(shè)A

(aij)mm,B

(bij)nn,X

(xij)mn,則:cs(

XB)

(BTcs(

AX

XB)n

(E

A

BT

E

)cs(

X

);mcs(

AX

)

(En

A)cs(

X

);證明：m

E

)cs(

X

).cs(

AX

)

cs(

AXE

)

(E

A)cs(

X

)Tncs(

AX

XB)

cs(

AX

)

cs(

XB)

(En

A

BT

Em

)cs(

X

)m

E

)cs(

X

)n

(