metodos de subespaciosrylov

8/14/2019 Metodos de Subespaciosrylov

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6/97

A Cmn

X = A1

A

AX = I

XA = I,

I

A

AX = XA.

AXA = A,

XAX = X,

(AX) = AX,

(XA) = XA

Ak+1X = Ak

k 0

M

M

(, )

H1=

Cn

H2=

Cm

A

A

Cmn

A

A: H1 H2

Hj

AX

XA

I


7/97

X

AX = PR(A) XA = PR(A) PU

U

R(M) :={Mv : v H1} H2

M : H1 H2

A#

A

A

A

k index(A)

A

0

index(A) = 0

A

A

index(A) 1

Ax=b,

x Cn = H1 b

Cm = H2 A x = A1b

A

b R(A)

N(A) ={v : Av = 0, v H1}

A

A{

}

A

A{

}

X A{

}

b

b AXb = minxH1

b Ax

(, )

Xb = minx,Ax=b

x b R(A)

X

A


8/97

Xb


9/97

Ac=r0 AXr0 A

X

A

A

T

lim T = O

AX A

XA = IT

AX A

X A

A

(T)< 1

1

A

b R(A)

T

T

I T


10/97

H

(, )

W

H

r

H

hMR r W rMR

hMR W, rMR :=r hMR W.

hOR

hOR W, rOR :=r hOR V,

V

W

W

A : H H

A : H1 H2

H

{u1, . . . , uk

}

H {u1, . . . , uk}

Uk =

u1 . . . uk

Uk = span{u1, . . . , uk} span{Uk} Ukg g =

1 . . . k Ck

1u1 + +kuk Uk

H Ck : (, uj)kj=1 A H

R(A) = {Av : v H} N(A) = {v : Av=0, v H}

A

A

(Ax, y) = (x, Ay)

x, y H


11/97

Ax =b

b H

x H

x0 0= r0 = b Ax0

{0} = C0 C1 C2 Cm Cm+1 H, dimCm=m,

xm = x0+ cm cm Cm

cm Cm

xm

hm:= Acm

rm=b Axm = r0 Acm=r0 hm.

hm r0 m

Wm := ACm.

rm hm

r0 Wm cm Cm

xm = x0 + cm

r0Wm cm Cm Acm=r0

hm= r0 rm=0

hm Wm r0 cm

A|Cm

c=hm

xm =x0+ cm rm = r0 hm

Cm N(A) cm

Cm N(A) = {0},

A

Cm A

m


12/97

{0} =W

0W

1 W

m1W

m .

dimWm =m

Wm1 = Wm

Wm

c

C

b AxMR = r AcMR = mincC

r Ac,

r=bAx0

hMR = AcMR

PW: H H W :=AC

hMR

= PWr, rMR

=r hMR

= (I PW)r W.

A

C

AC :=

(APC)

A

C

cMR

cMRmin = ACr

cMR = ACr+z z

(I ACA)u : u C

AC = (APC)

A

C

R(AC) = R(ACA) C

N(AC) = (AC) = W

ACAAC = AC

AAC = PW


13/97


14/97

N(A) C= {0}

R(ACA) = C

N(ACA) C= {0}

ACA

v H

v =

w+z

w R(ACA)

z N(ACA)

c C

c = d+z

d R(ACA)

c

d

C

z =c d N(ACA) C

R(ACA)

N(ACA)

C= C.

z N(ACA)

C

z=0

Az=0

Az W

Az W

z C

Az =0

ACAz =0

z

A(2)T,S

cMR

{z1, . . . , z} N(A) C Z=

z1 . . . z

cMRmin =cMR Z(ZcMR).


15/97

N(A) C

z N(A) C

cMRmin =cMR (z, c

MRL1)

(z, z) z.

cMR

cMR z | z N(A) C .

cMRmin

cMRmin = minzN(A)C

cMR z.

z =Zg

g C

mingC

cMR Zg

g = ZcMR = ZcMR

N(A) C

cMRmin

xMRmin

Z Z = PZ Z := C N(A)

cMRmin = (I PZ)cMR,

cMR z

z Z

cMR z2 = PZcMR z2 + (I PZ)cMR2

z=PZc

MR

x0+ cMR =x0+ ACr z z Z

xMRmin = (I PZ)(x0+ ACr) = (I PZ)xMR,

xMR

xMRmin = (I PZ)ACb+ (I PZ)(I ACA)x0


16/97

xMR

xMR = Ab+ z

z N(A)

cMR = Ar+z

z N(A)

hMR =PR(A)r

rMR =PN(A)r=PN(A)b

PR(A)r W

xMRmin

xMRmin = Ab

cMR

= A

r PN(A)x0

Ab x0+ C

Vm+1 := span

{r0

}+ Wm.

Vm+1 Acm

{v1, . . . , vm} Vm

cm Cm \Cm1 PVm

Vm

v1 = r0/, := r0,

vm+1= (I PVm)Acm(I PVm)Acm

(m= 1, 2, . . . ).


17/97

m

Acm

Vm

.

m

L:= min{m : Acm Vm}

L :=

H

L <

Cm :=

c1 c2 . . . cm

Vm+1 :=

v1 v2 . . . vm+1

m

A

ACm=Vm+1Hm= VmHm+ 0 . . . 0 m+1,mvm+1 ,

Hm =

j,km+1,mj,k=1

C(m+1)m

Hm :=Im 0

Hm Cmm Hm

Hm j,k = (Ack, vj) 1 k j m

k+1,k =(I PVk)Ack 0 k = L

Hm m

m < L

vL+1 = 0 m= L

HL

Vm+1 Vm+1 r0 = v1 =

Vm+1u

(m+1)

1

u

(m+1)

1

u

(m+1)

1

Cm+1

c Cm c = Cmy

y Cm

b Ax = r0 ACmy = Vm+1(u(m+1)1 Hmy) = u(m+1)1 Hmy2

2 Cm+1 cMRm = CmyMRm

u(m+1)1 HmyMRm 2= minyCm

u(m+1)1 Hmy2.

H

m

m

m < L

C

A

m < L

Cm N(A) = {0}

cMRm =CmyMRm =Cm

Hmu(m+1)1 = A

Cmr0

Hm

m

Hm = (HHmHm)

1HHm


18/97

Hm

Qm1Hm1=

Rm1

0

,

Qm1 Cmm QHm1Qm1=Im Rm1 C(m1)(m1)

Hm1

Qm1 0

0 1

Hm=

Qm1 0

0 1

Hm1 hm

0 m+1,m

=

Rm1 t0

0 m+1,m

.

Gm:=

Ik1 0 00 cm smeim

0 smeim cm

(cm, sm 0, c2m+ s2m = 1, m R)

Gm

Qm1 0

0 1

Hm=

Rm1 t0

0 0

Qm:=Gm

Qm1 00 1

Rm:=

Rm1 t0

.

cm:= ||||2 + 2m+1,m

, sm:= m+1,m||2 + 2m+1,m

,

m:= arg(m+1,m) arg() = arg(),

:=||2 + 2m+1,m eim .

Qm= Gm

Gm1 0

0 1

Gm2 O

O I2

G1 OO Im1

,

QmHm=

Rm0

.


19/97

m+1,m= 0 Rm

m < L

m+1,m = 0 m = L RL

= 0 = 0 cm sm

c2m+ s2m= 1

minyCm

u(m+1)1 Hmy2= minyCm

QHm

Qmu(m+1)1

Rm0

y

2

= minyCm

Qmu(m+1)1

Rm0

y

2

= minyCm

qm Rmy q

(m)m+1,1

2

,

[qm, q(m)m+1,1]

=Qmu(m+1)1 qm Cm Qm

yMRm =R1m qm

rMRm =|q(m)m+1,1|

Hm=QHm

Rm0

Hm=

R1m 0

Qm

m < L

ACm = Vm+1Hm=Vm+1QHm

Im0Rm=:

VmRm

m < L

m = L

Vm Wm

L

ACL=VLHL

u(L)1 HLyMRL 2= minyCm

u(L)1 HLy2,

yMRL c

MRL =VLy

MRL


20/97

L

HLy=u(L)1

A

L

cMRL = ACLr0 = CLH

1L u

(L)1

xMRL =x0+cMRL AxMRL =b

HL

yMRL =H1L u

(L)1 c

MRL =CLy

MRL r0 AcMRL =rMRL =0

AxMRL = Ax0+ AcMRL = Ax0+r0 = b

yMRL

HL

QL1HL=QL1 HL1 hL= RL1 t

0 = RL

= 0

L+1,L= 0 cL= 1 sL= 0 H1L =R

1L QL1

yMRL =R1L QL1u

(L)1 .

r0 = AcMRL WL

m

r0Wm= span{Ac1, . . . , Acm}

xMRm

Wm= Vm


21/97

m = L

ACL = VLHL

HL

WL= span{ACL} = span{VL} = VL,

Vm= span{r0} +Wm1 Vm= Wm r0Wm

r0 Wm Wm = ACm f Cm r0 = ACmf

r0= ACmf =Vm+1Hm+1f.

r0=Vm+1u(m+1)1

Hm+1f =u(m+1)1 .

Hm+1 Hm+1

m+1,m= 0 m= L

HLf =

HL1 hL

f =uL1.

f uL1

HL HL1

L1

HL HL

hL span{HL1} uL1

HL1 uL1

HL1

CL

HL

m

r0 AcMR = mincCm

r Ac = minyCm

u(m+1)1 Hmy2


22/97

rank(Hm)< m

HH

mH

m

Rm

Cm N(A) = {0}

Acm Wm1

Wm= Wm1

dimWm


23/97

g

HL1y = hL N(HL)

g=

g

1

=

HL1hL

1

.

rank(HL)< L rank

u

(L)1 HL

=

L

u(L)1 R(HL) rank(HL) =L 1 HL

N(A) CL N(HL) = span{g} N(A) CL =span{CLg} HL

QL1HL= QL1

HL1 hL

=

RL1 t

0

=

RL1 t

0 0

,

HL cL sL

minyLCL

u(L)1 HLyL2 = minyL1C

L1

C

QL1u(L)1

RL1 t0 0

yL1

2

=

minyL1CL1

C

qL1 RL1yL1 t q

(L1)L,1

2

,

qL1 q

(L1)L,1

=QL1u

(L)1 QL1

yL1 := R

1L1(qL1 t)

R1L1t=

HL1hL=g

HL1y=hL R

1L1qL1=y

MRL1

yMRL1 g

: C=

yMRL1

0 g : C

rMRL =|q(L1)L,1 | = rMRL1

L

HLy=u

(L)1

cMRL1 = CL1yMRL1 ACLr0+ N(A) CL,


24/97

(L 1)

rMRL = r

MRL1

cMRL1 (CL1g cL) : C

ACLr0=cMRL1

(z, cMRL1)

(z, z) z z =CL1g cL

g= HL1hL=R

1L1t

RL1 t

CL N(A)CL= span{CLg}

z =CLg=CL1g cL N(A) CL

Cm

N(A) =

{0

},

rank(Hm) =m,

dimWm=m,

HmHm Rm m

A

A

A

Rm

Rm

L


25/97

L

L= min{m : Acm Vm}= min{m : r0Wm}= min{m : Wm= Vm}= min{m : AxMRm =b}.

L

L= min{m : Acm Vm}= min{m : Acm Wm1}= min{m : Wm= Wm1}= min{m : dimCm>dimWm}= min{m : dimWm< m}= min{m : Cm N(A) = {0} }= min{m : rank(Hm)< m}= min{m : det(HHmHm) = 0}= min{m : rm,m= 0},

rm,m= Rm

C1 CL1 CL CL+1 W1 WL1 = WLWL+1= ACL+1 .

L

A

cL CL \ CL1

cL+1 CL+1\ CL

N(A) CL z

cL cL+1

CL+1\ CL cL z


26/97

Cm m > L + 1 Cm = span{c1, . . . , cL1, cL+1, . . . , cm} m =m 1

z

Z =

z1 . . . z

N(A)

span{z1, . . . , z}

H

Cm

c1=

100

, c2=

110

, c3 =

111

, . . . .

A =1 0 0

1 0 10 0 0

, b=

100

x0 =0

00

.

v1 =r0 =b Ac1 =

1 1 0

v1

v2=

0 1 0

A

1 10 1

0 0

=

1 00 1

0 0

1 1

1 1

,

g=

1

z =

01

0

c2

Ac3 v1 v2

A

1 10 1

0 1

=

1 00 1

0 0

1 1

1 0

,

cMR =

1 10 1

0 1

1 1

1 0

10

=

1 10 1

0 1

0

1

=

11

1


27/97

Ax = b

span{c1, c2, c3} = H

minC

cMR + z ,

= (z, cMR)

(z, z) = 1

cMR + z =

10

1

.

C

N(A)

Cm+ = Cm Z C Cm N(A) ={0} ZC N(A)

xMRmin = (I PZ)(x0+ ACmr0) = (I PZ)(ACmb+ (I ACmA)x0).

dimCm =m dimZ =

Cm Z

L

Lm

Cm N(A) ={0} m < L

m = L


28/97

A, b, x0,C

m:= 0, := 0, r0:=b Ax0, := r0, v1:=r0/, V1:=

v1

Z0, C0, Q0 H0

dimC> m+

Cm= span{Cm}, Z= span{Z}, Vm+1= span{Vm+1}

cC\ Cm Z

h := (Ac, vj)m+1

j=1

v := (I PVm+1)Ac, := v

t

:= Qmh ( := (Ac, v1) m= 0)

= 0

m:= m + 1

Cm:=

Cm1 c

, Vm+1:=

Vm v

, Hm:=

Hm1 h

0

Qm Rm

cMRm

= 0

g :=Rm1t

z :=Cmg c (z :=c m= 0)

z = (I PZ)z/(I PZ)z

Z+1:=

Z z

:= + 1

m:= m + 1

Cm:= Cm1 c , Hm:= Hm1 h , Rm := Rm1 t

0 cMRm :=CmRm

1Qm1u(m)1

Cm Z

xMRmin := (I PZ)(x0+cMRm ) PZ =ZZ


29/97

x, y H

(x, y) [0, /2]

cos (x, y) :=|(x, y)|xy .

x H U H

(x,U) := inf0=uU

(x, u),

cos (x,U) = sup0=uU

cos (x, u).

V

W

H

m:= min(dimV, dimW)

{j}mj=1 V W

cos j := max0=vV

max0=wW

|(v, w)|vw =:

|(vj, wj)|vjwj ,

v v1, . . . vj1 w w1, . . . wj1 V W

(V

,W

) :=m

U

H

PU

U

x H

(x,U) = (x, PUx)

sin (x,U) :=

1 cos2 (x,U)

PUx

=

x

cos (x,U),

(I PU)x = x sin (x,U).

m

hORm Wm, rORm :=r0 hORm Vm,

m

rORm Vm+1

Vm


30/97

V,W

H

r

H

h W

r h V

r W+ V

h

W V = {0}

PVW

H= W V

dimV= dimW

(V,W)< /2

hOR = PV

Wr

A

rMR = r hMR = (IW)r = r sin (r,W).

H= W V

PVW

= (PVPW), I PV

W = 1

cos (V,W)

rOR = r hOR = (I PVW

)r rcos (V,W)

.

cORm h

ORm = Ac

ORm

Hm Cmm

Hm m

Hm c

ORm

hORm = Ac

ORm

cORm =CmyORm y

ORm =H

1m u

(m)1 .


31/97

WL= VL PVLWL

= PWL

WL= WL1 VL

(VL,WL) = 0

m L

w1, . . . , wm {w1, . . . , wm}

Wm span{w1, . . . , wm1} = Wm1 wL := 0 dimWL =L

1

hMRm = PWmr0=m

j=1

(r0, wj)wj.

rMRm =r0 hMRm

rMRm 2 = (I PWm)r02 = r02 m

j=1

|(r0, wj)|2.

m

hMRm =m

j=1

(r0, wj)wj

=hMRm1+ (r0, wm)wm= hMRm1+ PWmr0 PWm1r0

=hMRm1+ PWm(r0 hMRm1) =hMRm1+ PWmrMRm1.

rMRm =r

MRm1 PWmr

MRm1= (I PWm)r

MRm1

rMRm = (I PWm)rMRm1 = sin (rMRm1,Wm)rMRm1.


32/97

m

m L

Qm = [q

(m)j,k ]

m+1j,k=1

Hm=

j,km+1,mj,k=1

sin (r0,Wm) = |q(m)m+1,1|

sin(rMRm1,Wm) =|q(m)m+1,1||q(m1)m,1 |

= m+1,m||2 + 2m+1,m

,

t

= Qm1hm hm= [j,m]

mj=1 Hm

{v1, . . . , vm+1}

Vm+1

r0=v1 u(m+1)1 Wm= ACm R(

Hm)

(r0,Wm) = (v1,Wm) = 2(u(m+1)1 ,R(

Hm)),

Cm+1

2

Vm+1Q

Hm Vm+1

v1 Qmu

(m+1)1

Qm Wm C

m+1

Cm

Cm+1

Cm

Im 00 0

Qmu(m+1)1

sin (r0,Wm) = sin 2(Qmu(m+1)1 ,C

m) =

=

Im+1

Im 00 0

Qmu

(m+1)1

2Qmu(m+1)1

2

= |q(m)m+1,1|,

sin (rMRm1,Wm) =rMRm rMRm1

= (I PWm)r0(I PWm1)r0

= sin (r0,Wm)

sin (r0,Wm1),

r0 Wm1 m

m = L

(r0,Wm) = 0 r0 Wm

rMRL =0

q

(m)m+1,1 =smeimq(m1)m,1 sm m


33/97

rMRm PWmrMRm1 Wm

rMR

m 2 =

rMR

m12

PW

m

rMR

m12 =

rMR

m12

|(r0, wm)

|2.

Wm1 r0 Wm1

m

wm = 0

cm=

1 s2m=

1 rMRm 2

rMRm1 =

|(wm, r0)|rMRm1

,

m

cm= 0

cos (Vm,Wm) = 0 dimWm = m

{w1, . . . , wm1, wm} wm := rMRm1/rMRm1 Vm

Vm Wm

|(w1, w1)| . . . |(wm1, w1)| |(wm, w1)|

|(w1, wm1)| . . . |(wm1, wm1)| |(wm, wm1)|

|(w1, wm)

| . . .

|(wm1, wm)

| |(wm, wm)

|

=

I 00 |(wm, wm)|

.

|(wm, wm)| =|(wm, rMRm1)|

rMRm1 =

|(wm, r0 PWm1r0)|rMRm1

=|(wm, r0)|

rMRm1 =cm.

m

(Vm,Wm) = (rMRm1,Wm).

VL WL

rMRL1 WL = WL1

GL

GL (VL,WL)

GL=

IL1 0 00 0 1

0 1 0


34/97

(rMRL1,WL)

m= L

r0 Vm Wm

(wm, r0) = 0

PR(A)r0Wm

PR(A)r0 = r0

m

r0 R(A) PR(A)r0 Wm m L

H

m+ 1, m+ 2, . . .

PR(A)r0 WL1

m

dim AVm+1= dimWm=m.

dimWm = m

m

c

Cm x0+ c

0 = A(b A(x0+c)) = Ar0 AAc.

Ar0 AWm AVm+1 =

Ar0+ AWm= A

Wm

dimWm= dim A

Wm dim AWm dimWm

dim AWm


35/97

(wm, r0) = 0 wm=rMRm1/(wm, r0)

(wm, wm) = 1

(wm, wj) = 0 j = 1, . . . , m 1

PVmWm

=m1j=1

(, wj)wj+ (, wm)wm.

hORm

hMRm = (P

VmWm

PWm)r0 =

(r0, rMRm1)

(wm, r0) (r0, wm)wm

=rMRm12 |(r0, wm)|2

(wm, r0) wm=

rMRm 2(wm, r0)

wm,

sm = sin (r

MRm1,Wm)

cm= cos (rMRm1,Wm)

rMRm =smrMRm1 =s1s2 smr0,

m

rMRm =cmrORm , rORm =s1s2 smr0/cm,

xMRm =s2mx

MRm1+ c

2mx

ORm +z, z Z

hMRm =s2mh

MRm1+ c

2mh

ORm

rMRm =s2mr

MRm1+ c

2mr

ORm .

m

1

rMRm 2xMRm =

mj=0

1

rORj 2xORj +z, z Z

1

rMRm 2hMRm =

mj=0

1

rORj 2hORj

1

rMRm 2rMRm =

mj=0

1

rORj 2rORj ,


36/97

1

rMRm

2

= 1

rMRm1

2

+ 1

rORm

2

=

m

j=0

1

rORj

2

,

j

cj= 0

rMRm rORm =hORm hMRm

rMRm =rORm +

rMRm 2(wm, r0)

wm.

rMR

m wm

rMRm 2 =

1 +rMRm 2(wm, r0)

rORm 2 =

rMRm12(wm, r0)

rORm 2,

hMRm hMRm1

hORm =hMRm +

rMRm 2(wm, r0)

1

(r0, wm)(hMRm hMRm1)

=hMRm +rMRm 2rMRm12

rMRm12|(r0, wm)|2 (h

MRm hMRm1)

=hMRm +s2mc2m

(hMRm hMRm1),

s2m+ c

2m= 1

hMRm =h

MRm1 m cm = 0

h

MR

m =

mj=0cj=0

2

m,jh

OR

j ,

m,0:=m=0c=0

s m,j :=cj

m=j+1c=0

s 1 j m cj= 0)


37/97

m,j := rMR

j rORj m

=j+1c=0

rMR rMR1

=rMRm rMRj

,

j= 0

rMRm = 0 rMRm 2

1 =s2m+ c2m=

rMRm 2

rMRm1

2

+rMRm 2

rORm

2

hMRm =hMRm1 cm= 0

hm

cm xm A|CmZ

z Z x0


38/97

Km(A, r0) := span{r0, Ar0, . . . , Am1r0}.

Cm= Km(A, r0)

Vm+1= span{r0} + ACm= span{r0} + AKm(A, r0) = Km+1(A, r0),

AVm= Vm+1Hm=VmHm+ m+1,mvm+1um.

A

A

A

A

b

x0 Ax = b r0 = b Ax0 R(A)

c

Ac=r0

Km(A, r0) = {q(A)r0 : q Pm1} (m= 1, 2, . . . ),

Pm m

Pm m

q Pm1 q(A)r0 = 0

m

mr0,A

mr0,A(A)r0 = 0 r0 A

Amr0 span{r0, Ar0, . . . , Am1r0},


39/97

L

L

L


40/97

Rm A

CmA

Cm = Km(A, r0) = span{Vm}

A

ACm

ACm

m

r0 m

AC

C

Km(A, r0) N(A) = {0}

KL1(A, r0) KL(A, r0)

r0

m

A

X

A

X

A

(AX

A)

2

= AX

A K

m(A, r0)R(AXA)

r

A

mA n n A

mA() =k0

i=1( i)ki,

i i= 1, . . . , A d:= k0

A

0 = 0 ki

i A

A Cnn

N Cnn

d

Nd = O

m < d

Nm = O

mN() =d

A Cnn

B Cmm

A O

O B

mA()mB()

AD

n n

A

ADAAD = AD,

ADA = AAD

Ad+1AD = AdAD,


41/97

d= index(A)

A

A

A = TC O

O N

T1,

T Cnn

C Crr

r = rank(Ad)

N C(nr)(nr)

d

A

AD = T

C1 O

O O

T1

AD =O

A

A

A

A Cnn

p

M=d+

i=1 ki

1 = deg(mA) 1 p(0) =p(0) = =p(d1)(0) = 0

p(j)(i) =(1)jj!

j+1i(j = 0, 1, . . . , ki 1 i= 1, 2, . . . , ).

p

p(A) = AD.

A

A = T

J O

O N

T1

J

N

J= diag(J1, . . . , Jh) N= diag(N1, . . . , Ng) Jk i Nj

Nj Cmm m d = k0 d d

N

N

d

mN() =

d

N

p

p(A) = T

p(J) 0

0 p(N)

T1

J1 = diag

J11, . . . , Jh

1

,

p(J) = diag (p(J1), . . . , p(Jh)) ,


42/97

p(Jk) = Jk

1

k = 1, . . . , h

p(N) = O

p

d

p(0) = p(0) =

=p(d1)

= 0

Jk =

i 1 0 . . . 00 i 1 0

0 0 i 10 0 . . . 0 i

C

mm

p(Jk) =

p(i)

p(i)

1!

p(i)

2! . . .

p(m1)(i)

(m1)!

0 p(i) p(i)

1! . . . p

(m1)(i)(m1)!

0 p(i) p(i)

1!

0 0 . . . 0 p(i)

Jk1 =

1i

12i

13i

. . . (1)m1

mi

0 1i

12i

0

12i

0 0 . . . 0 1i

.

i ki

ki p

(j)(i) = (1)jj!

j+1i

j = 0, 1, . . . , ki 1 i

A

p

p

M= deg(mA) 1

A = [ 0 10 0 ] mA() = 2

AD = O

p() = 0

0

p

M= deg(mA) 1

A Cnn

mA d= index(A)

A

p

p(A) = AD

M= deg(mA) 1


43/97

d = 0

A

p(A) = AD = A1

p

A

s() :=

1 p() mA degp deg(mA) 1

d > 0

p() 1 2 . . . d1 d . . . M

0 1 0 0 . . . 0 0 . . . 00 0 1 00 0 0 2

0 0 0 . . . (d 1)! 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p(j)(i) d!

(dj)!dji

M!(Mj)!

Mj1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

M+ 1

i

p()

p()

M

0 1 0 0 . . . 0 0 . . . 00 0 1 00 0 0 2

0 0 0 . . . (d 1)! 0 . . . 011

1 1 21 . . .

M11

121

0 1 21 . . . (M 1)M21

(1)(k11) (k11)!k11

0 0 0 (M1)!(Mk1)!

Mk1112

1 2 22 . . .

M12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)jj!

j+1i

d!(dj)!

dji(M1)!

(M1j)!M1ji

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)(k1) (k1)!k

0 0 (M1)!(Mk)!

Mk

= 0.


44/97

d 1

1 (1)j

j!j+1i

j+1

ij! = 0

0 1 0 0 . . . 0 0 . . . 00 0 1 00 0 0 1

0 0 0 . . . 1 0 . . . 0

1 1 21

31 . . .

M1

1 0 21 231 . . . (M 1)M1

(1)(k11) 0 0 0 M1k11

M1

1 2 22

32 . . .

M2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)j dj

d+1i

M1j

Mi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(

1)(k1) 0 0 M1k1M

.

i

ki kj i > j

0 1 0 0 . . . 0

0 0 1 0 . . . 0

1 1 21

31 . . . . . .

M1

1 2

3 . . . . . .

M

1 0 21 231 . . . . . . (M 1)M1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)j 0 0 j+11dj

d+11

M1j

M1

(1)j 0 0 j+1jdj

d+1j

M1j

Mj

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)(k11) 0 0M1k11

M1

(1)(k11) 0 0 M1k11

Mk11

,


45/97

j = max{i : 1 i , ki> j j 0 j < k1

d d

A

R(Ad) = R(Ad+1)

N(Ad) = N(Ad+1)

H= R(Ad) N(Ad)

A|R(Ad) A Ad

R(Ad)

x H

x=y+z y R(Ad)

z N(Ad)

ADx = (A|R(Ad))1y

C

A

R(Ad)

T

T

H= Cn

T

N

A

N(Ad)

H

H= Cn

A

A

CH

C

d

A

R(Ad)

N(Ad)

PR(Ad),N(Ad) = A

DA

R(ADA) = N(I ADA) = R(AD) = R(Ad)

N(ADA) = R(I ADA) = N(AD) = N(Ad).

j d

R(Adj)

d

A

j N0 0 j d

R(Adj) = R(Ad) N(Aj).

v R(Adj)

w H

v = Adjw

w=r+ s

r R(Ad)

s N(Ad)

v

v= Adjw= Adjr+ Adjs =:y+z,


46/97

y = Adjr R(Ad)

z = Adjs N(Ad)

v

Aj

z = Aj

Adj

s = Ad

s =0,

v R(Ad) N(Aj)

R(Adj) R(Ad) N(Aj)

v=y+z

y R(Ad)

z N(Aj) N(Ad)

r R(Ad)

s N(Ad)

y= Adjr

z = Adjs

v=y+ z = Adj(r+ s)

R(Adj)

A

mA

p

p(A) =O

r

A

mr,A

p p(A)r=0

degp

deg mr,A p mr,A

p() =s()mr,A() + r(),

deg r


47/97

r

A

mr,A=qc()

c(0) = 0

0 =mr,A(A)r=T

mr,A(C) O

O mr,A(N)

y

0

+ T

mr,A(C) O

O mr,A(N)

0

z

=T

mr,A(C)y

0

+ T

0

mr,A(N)z

,

mr,A(C)y = 0 mr,A(N)z = 0 my,C

mz,N mr,A N mz,N()

j

qc()

mz,N() =

j

j q

j < q

mr,A

p() :=jc()

mz,N() =q

Nqz =0

AqT

0

z

= T

0

Nqz

= 0,

T

0

z

N(Aq).

r R(Ad) N(Aq) = R(Adq)

q

z

N

j < q

T

0

z

N(Aj)

R(Adq)R(Adj)

q

j.

q

r

A

d = index(A)

Km(A, r) R(Adq) m

H

H= R(A0) R(A) R(A2) R(Ak)

A k


48/97

A

n

N(An) =N(An+1)

a

n

A

d:= a = n

d

H= R(Ad) N(Ad)

A|R(Ad) A R(Ad)

ADv = A|R(Ad)1y,

v

v = y+z

y R(Ad)

z N(Ad)

AD

q

r

A

mr,A

L= q+k

i=1

ni.

L

ADr

r

N(Ad)

d= index(A)

L= min{m : ADr Km(A, r)}.

L

r

A

L

KL(A, r)

ADr

KL(A, r) = {p(A)r : p PL1}

L 1

ADr =p(A)r

p(0) =p(0) = =p(q1) = 0

p(j)(i) =(

1)jj!

j+1i(j = 0, 1, . . . , ni 1 i= 1, 2, . . . , ),

A

P

T

A

r

A = TP

J O RJO N RN

O O R

PT1

r= TP

yz

0

,


49/97

J

i ni N q

degp = L 1

L > q

r0 =b Ax0 r0 N(Ad) = N(AD) ADr0 =0 Km(A, r0) N(Ad)

m

r0 A q r0 N(Aq)

r0= 0 d= 0 q > 0 R(Ad) N(Ad) ={0}

mr0,A() = q

L = q

rm= pm(A)r0 pm() := 1

qm1()

Pm qm1(A)r0=cm

cm Km(A, r0) m < L rL = rq = 0 L = q A

qr0 = 0

WL = AKL(A, r0) = span{Ar0, . . . , Aq1r0, Aqr0} = span{Ar0, . . . , Aq1r0} =AKL1(A, r0) = WL1

ADr

KL(A, r) A

AD

L r N(Ad)

KL(A, r) Ad

L

r

A

q

r

A

r=s + z

s R(Ad)

s = PR(Ad),N(Ad)r z = PN(Ad),R(Ad)r N(Ad)

KL(A, s) = ADAKL(A, r) = R(A

d) KL(A, r) KL(A, r), KL(A, z) = N(A

d) KL(A, r) = N(Aq) KL(A, r) KL(A, r) KL(A, s) KL(A, z) = KL(A, r).

q= 0

z = 0

N(Ad)KL(A, r) =

{0

}

m < L

q > 0

Km(A, PR(Ad),N(Ad)r) Km(A, r).

KL(A, r)

ADKL(A, r) = KL(A, s) =ADAKL(A, r) = R(A

d)KL(A, r)

AD

HL A


50/97

r H

q= index(r, A)

AVL=VLHL span{VL} = KL(A, r)

HL

q

v KL(A, r) v=VLy y CL

PR(Ad),N(Ad)v=VLHDL HLy R(Ad) KL(A, r),

PN(Ad),R(Ad)v=VL(I HDL HL)y N(Aq) KL(A, r) ADv=VLH

DL y.

HL

HqL N(Aq) KL(A, r) q >0 HL

HL

g(1) :=g= g1

=

H

L1hL1

=R1

L1t

1

.

g(2)

H2Lg

(2) =0

HLg(2) =0

HLg(2) =g(1)

= 0

index(HL) > 1

q 2

QL1

RL1 t

0 0 s

= f

, QL1g= f

y= s

.

QL1g

= 1

= 1

g(2) =

s

=

R1L1f +g

1

=

R1L1f

0

+g(1)

span{g(1), g(2)} = N(H2L)

{g(1), . . . , g(q)}

N(HqL) {VLg(1), . . . , V Lg(q)} N(Aq)KL(A, r) =

N(Ad) KL(A, r)

q

A

A

b

x0

Cm = Km(A, r0) r0 = b Ax0 xMRm xORm

q= index(r0, A) = 0 r0 R(Ad)

xMRL =xORL = A

Db+ (I ADA)x0=x0+ ADr0.


51/97

r0 Wm = AKm(A, r0)

p

m 1

r0 Ap(A)r0 =0 r() := 1 p()

r0 A

q= 0

q= 0

mr0,A() =L + L1

L1 + + 1+ 0,

0= 0 p() :=

1

0L1 +

L20

L1 + + 20

+10

.

r0 Ap(A)r0 = 0 p(A)r0 KL(A, r0) r0

AKL(A, r0)

m < L

r0 AKm(A, r0) = Wm

ADr0 KL(A, r0) cL= ADr0

rL= r0 AcL=r0 AADr0= (I AAD)r0=0,

r0 R(Ad) cL

ADr0

q= 0

b

Ad

x0 r0=b Ax0 R(Ad) b=0

x0=0

b R(Ad)

x0 Ax0

b

N(Ad)

PN(Ad),R(Ad)b= PN(Ad),R(Ad)Ax0

PN(Ad),R(Ad) = (I AAD)

r0 R(Ad)

Km(A, r0) R(Ad)

A

N(A) Km(A, r0) ={0} m L

A

r0

q= index(r0, A) = 0

AD


52/97

index(r0, A) = 0

m

L > m

ADr0

L

b

x

N(A) = N(A)

A Cnn

A = AD

AA = AA

R(A) = R(A)

R(A) N(A) Cn = R(A) N(A)

U

C Crr

r

A

A =UC OO OU

N(A) = N(A)

R(A) = N(A),


53/97

R(A) N(A)

index(A) 1

R(A) N(A) ={0}

index(A) 1

A = AD = A#

r0 R(A)

x0 q r0 A

xMRL =xORL =x0+ A

#r0 = A#b+ (I A#A)x0= Ab+ PN(A)x0.

q = 1

x0 L A

r0 =ADr0 KL(A, r0) = CL

AAr0 = PR(A)r0 AKL(A, r0) = WL.

m < L

PR(A)r0 Wm = AKm(A, r0)

cMRm = Ar0+z

z

N(A)

xMRm = Ab+ z

z N(A).

L

z

N(A)KL(A, r0) = {0} cMRL = A

#r0 = Ar0

xMRL = A#b+ (I A#A)x0 = Ab+ PN(A)x0,

x0 R(A) N(A)

Ab

A

b

x0

A =

1 0 01 0 1

0 0 1

R33.


54/97

A2 = A

index(A) = 1

A = AD = A#

A =2/3 1/3 1/30 0 0

1/3 1/3 2/3

N(A) = span

0 1 0

b =

1 1 1

A

cMR2 = c

MR1 +z

z K2(A, b) N(A) = N(A) cMR1 =

1 1 1

111 A(

111) =

111

101 ,

= 1

1

Ab=

Ar0 =

4/3 0 2/3

1/

3

xMR2 = cMR2 = xMR1 z11

1

01

0

,

= 1

cMR2 = A

K2(A,b)b = ADb

b=

1 0 0

R(A)

cMR1 =

1/2 0 0

ADb=

1 1 0

Ab=

2/3 0 1/3

A

index(A) = 1

A

index(A) = 1

index(A) 1

A

index(A) 1

R(A) N(A) = H

R(A) N(A) = {0}

A#


55/97

A

N(A) ={0} R(A) = H index(A) = 0

A

index(A) 1

d

A

R(A) = R(Ad) N(Ad1).

N(A)

0

R(Ad)

d = 1

d

N(Ad1)

N(A) =

{0

}

d = 1

d > 1

N(A) N(Ad1)

A

index(A) = 1

PN(A) PN(A),R(A)

index(A) = 1

b R(A)

xMRL =xORL =x0+ A

#r0 = A#b+ (I A#A)x0= A#b+ PN(A),R(A)x0,

AxMRL = AxORL =b

A#

x0 m r0 Wm= AKm(A, r0)

r0 R(Ad)

d = index(A)

r0 = b Ax0 x0

x0 = 0 b R(Ad) x0 Ax0 R(Ad)

x0 = s + t s R(Ad)

t

N(Ad)

Ax0 =As+ At=As

At= 0

t N(Ad)

d= 1

d= 0

x0 index(A) 1 b R(A) index(A) 1

b R(A)

x0

x0 index(A) 1 b R(A)


56/97

A = AD

Ar= ADr

r

A

r R(A)

r = Ay

y Cn

AAA = A

r= Ay= AAAy= AAr= AADr,

r R(Ad

)

d= index(A)

r R(A)

= N(A

) =N(A)

(I AAD)r=r AADr=r AAr=r

r N(Ad)

r

d= index(A)

PR(A)r=PR(Ad),N(Ad)r PN(A)r=PN(Ad),R(Ad)r

s := PR(A)r R(Ad) t := PN(A)r N(Ad)

r = s+t

R(Ad) N(Ad) R(Ad) N(Ad) = Cn

PR(Ad),N(Ad) = AAD

r

R(Ad)

N(Ad)

I AAD = PN(Ad),R(Ad)

ADr0

x0+ A

Dr0

cMRL =A

Dr

Ac =r

L

AADr= AAr

AD A r N(A)

cMR

Ac =r

PR(A)r AcMR = 0

PR(A)r= AAr= AcMR

cMRL = A

Dr

AADr = AAr

cMR = ADr

L

AcMR = AADr =AAr=PR(A)r c

MR


57/97

PR(Ad),N(Ad)= A

DA

H

R(Ad) N(Ad)

AD = AAAD = ADAA.

PR(Ad),N(Ad) = ADA = AAD

PR(A) = AA

PR(A) = AA

AD =PR(Ad),N(Ad)A

=APR(Ad),N(Ad)

PR(Ad),N(Ad)

PR(Ad),N(Ad) = ADA

PR(Ad)

UC R(A

d) R(A)

UC

W =

UC UW

R(A)

H = R(A) N(A)

R(A)

N(A)

UC UW Y

H

Y

N(A)

UY :=

UW Y

H= span{UC} span{UY}

H

UC R(Ad)

N(Ad)

UY

Z

N(A) N(Ad)

UZ = UV Z N(Ad) R(Ad) N(Ad) UC

UZ

UC UZ

=

UC UV Z

H

V :=

UC UV

H= span{V} span{Z}

H

span{Z}

A

R(A)

V

R(A)

A

UC UY

=

UC UY C O

O NY

A UC UZ= UC UZ C OO NZ

AD

UC UW Y

=

UC UW Y C1 O OO O O

O O O

= UCC1 O O

AD

UC UV Z

=

UC UV Z C1 O OO O O

O O O

= UCC1 O O


58/97

AA

R(A)

AA

W Y

=

W O

=

UC UW Y I O OO I O

O O O

,

ADAA

UC UW Y

= AD

UC UW Y I O OO I O

O O O

=

UCC1 O O I O O

O I O

O O O= UCC1 O O= AD UC UW Y .

AA =PR(A)

AA

V Z

=

UC UV O

AAAD

UC UV Z

= AA

UC UV Z

C1 O O

O O O

O O O

=

UC UV O

C1 O OO O OO O O

= UCC1 O O= AD UC UV Z ,

A

A

UC UV Z

=

UC UW Y

C O O

O M O

O O O

,

C Crr M Css r= rank(Ad) s= rank(A)r

span{Z} = N(A)

span{Y} = N(A)

R(Ad)

A

AUC=UCC (2, 1)

A

N(Ad)

AUV =UWM+ YB (1, 2)

UW Y UC


59/97

R

(Ad

) N

(Ad

)

b

x0 L c

MRL KL(A, r0) r0=b Ax0

PR(Ad)cMRL = A

DAcMRL = ADr0 = PR(Ad)A

r0.

L

r0 Ac = PR(Ad)r0 APR(Ad)c + PN(Ad)r0 APN(Ad)c

c KL(A, r0)

A

d

PR(Ad

)r0 APR(Ad

)c = 0

PR(Ad

)c= AD

r0K

L(A, r0) R

(Ad

)

AD = AAAD R(Ad) R(A)

AD = ADAA N(A) N(Ad)

R(AD) = R(Ad)

AA = PR(A)

R(I AA) = N(A) N(Ad)

AD(I AA) = O

R(AD) = R(Ad) R(A) = R(AA)

R(IAA) = N(A) N(Ad) = N(AD)

A

AD = ADAA = AAAD.

R(Ad) R(A)

AAAD = AAAD

A

R(Ad)

v H

AAv AAv

N(Ad)

N(A) N(Ad)

A

R(Ad) R(A) N(A) N(Ad),

R(Ad) N(Ad), d= index(A),


60/97

N(A) = N(Ad)

dimN(Ad) = dimN(A) = dimN(A)

N(A)

N(Ad)

N(A) = N(Ad)

d = index(A) 1

N(A) = R(A) R(A) N(A) = N(A),

d = 1

R(Ad) = R(A)

d 1

R(A) = N(A) N(A) R(A) = R(A).

A

TA = T

C OO N

C Crr

r = rank(Ad) d = index(A)

N C(nr)(nr)

d

C

N

A

Y =

y1 . . . yt

N(A)

R(A)

U

r

Ad

T

T = U X Y X Y N(A

d)

N(A

)N

(Ad

)

Y

U

V =

v1 . . . vs

s= n r t

U V

R(A)

U V Y

H

U V Y

U V Y

= I

y N(A)

y=Yg

g Ct

UY = O

0=TAy= (A T)y=

CH O

O NH

U

X

Y

Yg=

CH O

O NH

0

XYgYYg

.

g

t

X Y

Y

NH

t

A

S=

SXSY

N(NH)

N

Y

SY = Y

Y

ts

SX=XY

X

H


61/97

Y

N

Y

SY = I

t

N(NH

)

JN N = TNJNTN

1

t

t

TN

t

N

N

A

t

t

Y

N(A) N(Ad) = span{X, Y}

A

z =

X Y gX

gY

,

g=

gXgY

N

{g1, . . . , gt} N(N)

gj =

g

(j)X

g(j)Y

j = 1, . . . , t

R(Ad) R(A) = N(A)

U

z N(A)

0 = U

X Y

gj =

UX O

gj = U

Xg(j)X

j = 1, . . . , t

tr

X

g(j)X G Cst

UXG= O

tr+ts

s

X =

x1 . . . xs

X = YSHX

X

R(Ad) N(Ad)

UX= O

F Cts

GF = Is

X

F

G

GFG= F

rank G=s

U V Y

H

X

X = YSHX+V +UM M Crs SX = X

Y

UX= O

M= O

UXG= O

MG= O

H(I GG ) | H Crs .

MG = O

GG = I

G

rank G=s


62/97

N

G =

g1 . . . gt N(N) s t G s

s = 1

G

g

F =g/(g, g)2 g=0 N(A)

N(Ad)

s = 1

A

s > 1

dimN(Ad) dimN(A) + 2

N

t < s

G Cst

s

t s

t < s

A

d

A

N(Ad)

A

A R44

A

3 3

N =

0 1 00 0 1

0 0 0

.

C = 1 index(A) = 3 r =

rank(A3) = 1 N(A) t= 1

s = n r t = 2

SHX =

0 0

SY =

1

Y

Y =

1000

, U=

0100

V =

0 00 01 00 1

.

N

G = 1

0

M

MG= O

M =

0 1

X=YSHX+ V + UM =

1000

0 0 +

0 00 01 00 1

+

0100

0 1=

0 00 11 00 1

.


63/97

A =

0 0 0 0

1 1 0 10 0 0 11 0 0 0

KMR : H H

r L

A

KMRr=c MRL = AKL(A,r)r.

H

KMR

H

L

r

A

r

KMR

A

KMR = A1

KMR = AD

R(Ad)

KMR

R(Aindex(A))

A =0 1 00 0 1

0 0 0

r=

0 1

= 0

L= index(r, A) = 2

H2 =

0 01 0

, K2(A, r) = span{

01

0

,

10

0

},

2

cMRL1=cMR1 =0

K2(A, r)

cMRL =c

MR2 =0

= 0

L = index(r, A) = 3

K3(A, r) = H = R3

cMRL =cMR3 =

0 0 01 0 1

0 1 0

01

=

00

1

||

cMRL1

cMR2 =

1/2 0 1

3,2 = (I PV2)A2r =O(2)


64/97

KMR

m

Vm

xm

rm = b Axm = r0 Aqm1(A)r0 = pm(A)r0 Km(A, r0)

Cj = Kj(A, rm) j = 1, 2, . . .

index(rm, A) = 0

index(r0, A) = 0

pm(0) = 1

m

L

r0 R(Aindex(A))

r0 R(Aindex(A)) index(r0, A) = 0

R(Aindex(A))

index(A) = 1

A

R(Ad)

Aq

q = index(r0, A)

q

d= index(A)

d= 1

A

A b R(A)

R(A) N(A)

q

n n

q

d

n

Ak

A


65/97


66/97

X(t)

S

{X(t) | t T}

(,F, Pr {})

S

F

Pr {} : F [0, 1]

t

T

X(t, ) = X(t)

{ | X(t, ) = i} ={X(t) = i}

i S

Pr {X(t) =i}

Pr {X(s) =x(s)}

x : T S

S ={1, 2, . . . }

T= [0, )

s T

s, s + t T

t

X(t)

t X(t, ) = X(t)


67/97

s T

s

i

j

X(s) =j

X(t) =i

t < s

s

j

s

Pr {X(s) =j}

Pr {X(t) =i, X(s) =j}

i

t

j

s

Pr {X(s) =j| X(t) =i} = Pr {X(t) =i, X(s) =j} / Pr {X(t) =i}

t < s

j

s

i

t

t

(t) :=

Pr {X(t) =i}iS

=:

i(t)iS

.

i

i(t) (t) i

t

t T (t)

S

0 i(t) 1 iS

i(t) = 1,

1 :=

1 . . .1

1

(t) 0 (t)1 = 1.

(t)

t T

>

0

X(t)

S ={0, 1, 2, . . . }


68/97


69/97

i

i

:= 0 P

, 0, 0

P

(X)0 X

S= {1, 2, . . . , n}

X(t)

S

s, t 0

i,j,x(u)

Pr {X(s + t) =j| X(s) =i, X(u) =x(u),0 u s}= Pr {X(s + t) =j| X(s) =i} .

i,j,i1, . . . , ik S 0 s1 < < sk < s

t >0

Pr {X(s + t) =j| X(s) =i, X(s1) =i1, . . . , X (sk) =ik}= Pr {X(s + t) =j| X(s) =i} .

s,t,h T

0 s < t < h

i ,j,k S

Pr {X(s) =i, X(h) =k| X(t) =j}= Pr {X(s) =i | X(t) =j} Pr {X(h) =k| X(t) =j} .

Pr {X(s + t) =j| X(s) =i}

s

X(t)

X(s)

X(h)


70/97

pi,j(t) := Pr {X(s + t) =j| X(s) =i} = Pr {X(t) =j| X(0) =i} , i, j S,

P(t) = [pi,j(t)]i,jS,

X(t)

i S

pi,j(t)

jS

S

jS

pi,j(t) = 1,

0

1

P(t) O, P(t)1 =1 .

P(t)

P(t) O

P(t)1 1

>0

P()

X(t)

(0) =

i(0)iS

i(0) = Pr {X(0) =i}

(t) = (0)P(t).

t

P(t)

(0)

t >0

t= 0

P(t) =pi,j(t)

i,jS

i, j

i =j

t >0

pi,j(t)> 0


71/97

0

P(t) = tT

:=/(1 )

X(t)

(t) = (0) =

P(t + s) = P(t)P(s),

P(t)

:= limt

(t) = (0) limt

P(t),

(0)

> 0 (0)

(0)

limt

P(t)

limt

P(t) =1 =: .

P(t) =

t T

limt0

P(t) = P(0) = I.

P(t)


72/97

1

A =

ai,jni,j=1

Rnn, 0 ai,j 1,n

j=1

ai,j 1 i j

A

A

A O

A1 =1

A1 1 .

1

1

(A)

A

A Rnn

(A) 1

A

(A) = 1

A

1

A1 = 1

1

A

A

Q Cnn

n > 1

T

r

0< r < n

TQT=

A B

O C

ACrr

B Cr(nr)

C C(nr)(nr)

O C(nr)r

Q


73/97

Q

Cnn

i, j

k1 =

i, k2, k3, . . . , km1, km = j qk1,k2, qk2,k3 , . . . , q km1,km

A, B Cnn

A =

ai,j

n

i,j=1

B =

bi,j

n

i,j=1

i

j

i

=j

ai,j = 0 bi,j = 0,

A

B

A Cnn

A

A

P Rnn

1

P

1 P= > 0

1

P

P

P

(P) = 1

TPT =

P1,1 O . . . OO P2,2 . . . O

O . . . O Pm,m

,

T Rnn

Pj,j

(Pj,j) = 1

1


74/97

T R

nn

limk

Tk

P =

pi,j

ni,j=1

P

P

1

1

pi,i> 0 i= 1, . . . , n

=

=

.

A Rnn

A =I

B, >0, B

O,

(B)

M

A

M

=(B)

M

A

B/

M

A

index(A) 1

P(t) Rnn

t 0

P(0) = I

P(s + t) = P(s)P(t)

s, t 0


75/97

{P(t) | t 0}

I

P(t)

P(t) O

t 0

P(t)1 1

t 0

limh0

P(t + h) = P(t)

t, h 0

limh0

P(h) = I.

P(t)

t 0

P(t)

1

P(t)1 = 1

t 0

P(t)

P(t)

P(t) =

pi,j(t)

n

i,j=1

P(t)1

t

P(t)

t >0

t 0

i= 1, . . . , n

0

nj=1j=i

pi,j(t) 1 pi,i(t) t 0

limh0pi,i(h) = 1

P(t)

pi,i(t)> 0 t 0

pi,i(t) = 1 t >0 pi,i(t) = 1 t 0

pi,j(t) = 0 j=i t 0


76/97

|pi,j(t+h) pi,j(t)| 1 pi,i(|h|) t 0 h < t

P(t)

P(t)

[0, )

P(t) = d

d tP(t) = QP(t) = P(t)Q,

Q := P(0)

P(0) = I

P(t) =etQ =k=0

tk

k!Qk.

Q = P(0)

P(t)

P(t) = P(t)Q

P(t) = QP(t)

S() :=

0P(t) d t

S()

>0

I 1S() < 1

P(t)

c ( 1

2, 1)

> 0

pi,i(t) > c i

0 < t <

pi,i(0) = 1 pi,i(t)

t

i

i

I 1S() = nmaxi=1

n

j=1

i,j 1

0

pi,j(t) d t=

nmaxi=1

1 1

0

pi,i(t) d t

+1j=i

0

pi,j(t) d t

,

1

0

pi,i(t) d t 1

1

0

j=i

pi,j(t) d t 1

0

(1 pi,i(t)) d t = 1 1

0

pi,i(t) d t.


77/97

I 1S() n

maxi=1

2

1 1

0pi,i(t) d t

nmaxi=1

2 (1 c)< 1

c

pi,i(t)> c t (0, )

0pi,i(t) d t c 1 c < 12

c ( 12

, 1)

1

h(P(h) I)

0

P(t) d t= 1

h

0

P(t + h) d t

0

P(t) d t

.

s= t + h

h

h

S()

1

h(P(h) I) =

1

h

+h

P(t) d t 1h

h0

P(t) d t

0

P(t) d t

1.

h 0

P(0)

Q :=

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