Markov Chain

article version

180705 2

. Cyclic Decomposition Theorem .1

Cyclic Decomposition Theorem 2 ()

. motivation.2

[ I, 8.58.7] Cyclic Decomposition Theorem , Jordan canonical form ,

Markov chain 1

.

article Markov chain subject

. Markov chain

application . article Markov chain

,

() (genotype frequency),

() Google ranking system

. , Markov chain Perron-

Frobenius Theory () .

article Jordan canonical form([I, 238 (j) t ] [ I, 239 i ]) . , Cyclic Decomposition The-orem . , Jordan canonical form ,

article () F =C.1, , [3] Cyclic Decomposition Theorem .

1 1- , 2 2-

.2 [II, 12] .

1

1 section version

180705 componentwise .

1.1 Ck =(c(k)ij

)Mm,n(C), i, j lim

kc(k)ij

, {Ck}k=0 ,

limk

Ck =(limk

c(k)ij

) . Mm,n(C) = Cmn identify

.

2 (1.2,

1.3 1.6).

1.2 (Exponential map) exp : Mn,n(C) Mn,n(C)

exp(A) =k=0

1

k !Ak, (AMn,n(C))

.

.3

, , M. Artin [1, p. 119].

1.3 (Eigen-vector motivation) () A=

(2 1

3 4

)Mn,n(R)

( ) eigen-vector

. Ae1 = (2, 3)t Ae2 = (1, 4)

t, LA 1

S ={ae1+ be2 | a, b 0

} cone AS =

{a(2, 3)

t+ b(1, 4)

t | a, b 0}

. A2e1 = (7, 18)t A2e2 = (6, 19)

t, LA AS cone

A2S ={a(7, 18)

t+ b(6, 19)

t | a, b 0}. ,

cone sequence

S AS A2S A3S A4S

.

3Exponential map Lie group Lie algebra.

2

() cone half-line .4 , =

m=0 AmS

1 () half-line. ,

A = A(

m=0 AmS)=

m=0 Am+1S =

, LA half-line . X (non-

zero) vector, AX = X A eigen-value R. > 0.

() = 5, (1, 3)t half-line

([I, 7.2.1 ). (A eigen-value 1.)

Perron-Frobenius Theory( 7).

() eigen-value = 5 dominant eigen-

value.

1.4 A Mn,n(C) , A eigen-value(in C) maximum ab-solute value A dominant eigen-value. ( A

dominant eigen-value.) Dominant eigen-value

eigen-vector dominant eigen-vector . (

principal eigen-vector.5)

Numerical Linear Algebra Matrix Computation

, , size eigen-value( ) eigen-

vector( ) . n 10( 230)

characteristic polynomial 10 root

.

eigen-vector dominant eigen-vector

. Matrix Computation ,

(!), dominant eigen-vector.6

4 , A Am

. (, eigen-vector , eigen-

vector .)5 Principal Component Analysis(PCA) principal.6Matrix Computation Golub-Van Loan[4] Meyer[5]

. .

3

A Mn,n(C) dominant eigen-vector , 1.3 . , X Cn , lim

mAmX ( )

.7 , , 1.3 , m Am

AmX , .

limm

AmXAmX ?

A,A2, A3, . . . step n2-

. AX, A(AX), A(A2X), . . . step n-

. limm

Am limm

AmX .

n, n2 n significant.

1.5 Numerical Linear Algebra, ,

Linear Algebra.

() A Mn,n(C) . det(A) = 0, A() 230

.

, .

() A Mn,n(C) diagonalizable . A(t) ([I, 7.2.18] ).

AMn,n(C) unique dominant eigen-value.

1.6 A Mn,n(C), X Cn , lim

mAmX

AmX A dominant eigen-vector.

() m AmX = 0 ( A invertible, X = 0 OK). A diagonalizable. , AXi = iXi i C Xi Cn,

U1AU = D = diag(1, . . . , n), AmX = UDmU1X

. U = (X1, . . . , Xn)([I, 7.2.13] ).

A Mn,n(C) unique dominant eigen-value 1 , 1 > 0. j = 1 1 = j , . U = (uij), U

1X = (yj), y1 = 0.71.3 , 1 ( ) half-line ,

limm Am.

4

() limm

AmXAmX k- , 1 > 0 unique domi-

nant eigen-value 1 = j(, j = 1),j ukj

mj yj

i |

j uijmj yj |2

=m1

j ukj(j1

)myj

m1

i

j uij

(j1

)myj2 y1

i |ui1y1|2uk1

. limm

AmXAmX [U ]

1 =X1(

i |ui1y1|2 = 0

?). , limm

AmXAmX A dominant eigen-vector.

() . ()

. , 1.5

1.7 (), .

() () essential assumption 1 > 0 . (

() 1 > 0 .)

.

positive real number zero.

() A diagonalizable , A Jordan canonical form

. Detail.8

() () () L2-norm L1-

norm . A Markov matrix

X provability vector( 2.2 4.5() ),

m AmX L1-norm 1, limm

AmX , A

dominant eigen-vector( 4.9 ).9

1.7 1.6(), y1 = 0 X / X2, . . . , Xn.10

, limm

AmXAmX

? Google Ranking System( n 10) A52X

( 6).

8() . 3.2 .9 [6] L1-norm.

10 Hint: Uei =Xi.

5

terminology .

1.8 A Mn,n(C), X Cn , AX, A(AX), A(A2X), . . . lim

mAmX

AmX () , A dominant eigen-vector()

power method .

1.9 () , power method.

() , n 230 , A X 1.6()

. power method

, limm

AmXAmX . , A X

230 . ( idea 800 Google Ranking System. 6.)

() Power method

.11

A unique dominant eigen-value positive real number

, . , , unique dominanteigen-value positive real number square matrix

. Positive Markov matrix( 2.1 2.2 )

. ( , non-negative matrix( 2.1 )

.12 non-negative matrix dominant eigen-vector

Perron-Frobenius Theory.13)

, () Cn- norm 180705

. X = (x1, . . . , xn)t Cn, X norm()

X=|x1|2 + + |xn|2

.14 X,Y Cn X Y .

11. = .127 Meyer[5] . Non-negative case.13, power method computer , power method

Perron-Frobenius Theory . Perron-Frobenius Theorem

20 . non-negative matrix

, .14 Cn R2n naturally identify.

6

2 A Toy Example

section version

180102 Markov matrix, toy example.

2.1 C = (cij) Mm,n(C) , [cij > 0 for all i, j ], C > 0 C positive .15 [cij 0 for all i, j ],C 0 C non-negative . , X Cn ,X > 0 X 0.

2.2 () X Cn , X 0 X 1, X probability vector . A = (aij) Mm,n(C) , A column probability vector(, A column sum 1),

A (left) Markov matrix( A 0).16

() At Markov matrix, A right Markov matrix.

Markov matrix. :

2.3 (Right) Markov matrix 1 eigen-value .

: A(t) = At(t)([I, 7.6.24]), right

. . A

right Markov, A row sum 1,

A (1, . . . , 1)t = 1(1, . . . , 1)t

. , A Markov, (AI) row zero vector, (A I) row, det(1A I) = 0.

(2 2)- example , Markov matrix (2 2)- . toy example [3] ( )

. ( toy example idea 5 6(Random Surfer

Model).)

15 positive definite matrix (Linear Algebra)

. [I, , 15.3] .16Markov matrix stochastic matrix, probability matrix, transition matrix

. Probability vector stochastic vector.

7

2.4 () (city) (suburbs)

(probability vector) X = (c0, s0)t (, , c0, s0 0

c0+s0 = 1).17 ,

A=

(0.90 0.02

0.10 0.98

)=

(city city suburbs city

city suburbs suburbs suburbs

)

. , , 1 city 90%

city , city 10% suburbs .

, 1 (c1, s1)t(

c1

s1

)=

(0.90c0+0.02s0

0.10c0+0.98s0

)=

(0.90 0.02

0.10 0.98

)(c0

s0

)= AX

. , 2 (c2, s2)t(

c2

s2

)= A

(c1

s1

)= A2X

, m- AmX.18

() A,

U1AU =

(16

16

56

16

)1(0.90 0.02

0.10 0.98

)(16

16

56

16

)=

(1 0

0 0.88

)=D

. ,

Am =UDmU1 =

(1+5(0.88)m

61(0.88)m

655(0.88)m

65+(0.88)m

6

), lim

mAm =

(16

16

56

56

)

,

AmX =

(1+5(0.88)m

6 c0+1(0.88)m

6 s055(0.88)m

6 c0+5+(0.88)m

6 s0

), lim

mAmX =

(1656

)

().

17c for city, s for suburbs.18 , mathematical model A constant,() () ( ) . ,

, 1.

8

() A , suburbs city

. (16 ,

56

)t equilibrium()

. city 0 . ( suburbs

2% city, suburbs .)

() A positive Markov matrix. 1 A unique dominant

eigen-value, dominant eigen-vector U column(16 ,

56

)t. (Dominant eigen-vector probability vector

(16 ,

56

)t.)

, () limm

AmX A dominant eigen-vector

1.6.

() , limm

Am column A dominant eigen-vector

.

() equilibrium(16 ,

56

)t

X = (c0, s0)t .

c0 = 0 s0 = 0 equilibrium.

() () Am AmX, 0.88m 0

, .

toy example [ I, 7 ].

toy example ( 4.8 4.9

).

9

3 Preliminary Resultssection version

180102, F = C, A Mn,n(C) . A

characteristic polynomial

A(t) = (t1)e1(t2)e2 (tk)ek

(, ei 1, i distinct). ei eigen-value i multiplicity, ei = multA(i). A -eigen-space

EA = ker (AI) = {X Cn |AX = X}

([I, 7.6.20] ).

, article !

3.1 Prove or disprove : dimEA = multA().19

Jordan canonical form .20 A similar Jordan

canonical form J i-Jordan block Ji Mei,ei(C) block diag-onal matrix. Ji i-Jordan -block Jij Mrij ,rij (C) block diagonal matrix. ,

A J = diag(J1, J2, . . . , Jk), Ji = diag(Ji1, Ji2, . . . , Jihi)

.21 -Jordan -block Jij

K = (),

( 1

0

), . . . ,

1

1 0

0 1

. , A diagonalizable, A Jordan canonical form

diag(1Ie1 , 2Ie2 , . . . , kIek).

19 A eigen-value, dimEA =0= multA().20, Cyclic Decomposition Theorem OK.21Jordan block, Jordan -block.

10

Jordan canonical form (upper-)triangular matrix, triangular

matrix diagonal matrix .

Jordan canonical form

. , m Jm

. , J A Jordan canonical form , Jm ,

{Am}m=0.

3.2 K = I+N Mr,r(F ) , N2, N3, . . . . , m, m r1,

(K)m =

m mm1(m2

)m2

(m3

)m3

(m

r1)mr+1

m mm1(m2

)m2

(m

r2)mr+2

. . .. . .

. . ....

0. . .

. . ....

m mm1

m

.22 (, , m < i binomial coefficient

(mi

)= 0

. , = 0 m < r 1 . = 0, (K)

m =Nm.23)

, , .

3.3 Ak Mn,n(C), B Mm,n(C), C Mn,r(C) U Mn,n(C), .

() {Ak}k=0 , limk

Ak = L . , {BAk}k=0{AkC}k=0. lim

k(BAk) =BL lim

k(AkC) =LC.

() {U1AkU}k=0 {Ak}k=0 . lim

kAk =L, lim

k(U1AkU) =U

1LU .

22Hint: I N commute , (I)N =N(I) (I+N)m 2-

.23 N nilpotent.

11

.

3.4 D={C

|| < 1 or = 1}.24 . , {m}m=0

D. {Am}m=0.

3.5 AMn,n(C) , {Am}m=0

(i) C A eigen-value, D,(ii) 1 A eigen-value, dimEA1 = multA(1)(, A 1-Jordan

block ).

: J A Jordan canonical form , 3.3()

, {Am}m=0 {Jm}m=0 . 3.2 (K)

m. (.)

3.1 ,

.

3.6 A eigen-value,

(1) dimEA = multA().

(2) A -Jordan block I.

.

Linear Algebra A eigen-value 1, . . . , k

. , Numerical Linear Algebra 3.5

eigen-value () . Eigen-value

.

3.7 A= (aij)Mn,n(C),

i(A) =n

j=1 |aij |, j(A) =n

i=1 |aij |

. A row sum column sum

(A) = max {i(A) | 1 i n}, (A) = max {j(A) | 1 j n}

.

24D disk.

12

simple, .

3.8 (Gershgorins Disk Theorem, 1931) A = (aij) Mn,n(C)eigen-value eigen-vector X = (xi) Cn . |xk|= max

{|xi|

1 i n} ,25 |akk| k(A)|akk|

.26

: AX = X,n

j=1 akjxj = xk, j =k akjxj = xkakkxk

. , xk = 0,

|akk| =j =k akj xjxk j =k |akj | = k(A)|akk|

.27

3.7 row sum column sum.

3.9 AMn,n(C) eigen-value, || min{(A), (A)}.

: Gershgorins Disk Theorem,

|| |akk|+ |akk| (k(A)|akk|

)+ |akk| = k(A) (A)

. At eigen-value( ?),

|| (At) = (A)

.

Gershgorins Disk Theorem 3.9

. , A eigen-value A similarity class invariant

, (A), (A) similarity class invariant . , , A eigen-value A n2-.

25 k.26 akk k(A)|akk| disk Gershgorins disk.27 1931 . (

A. Markov(18561922) .) historical comment

Meyer[5, p. 497] .

13

Positive matrix eigen-value a technical lemma

. motivation .28

3.10 A Mn,n(C) , A > 0 . A || = (A)eigen-value .29 , .

() = (A)> 0.

() dimEA = 1. EA =

(1, 1, . . . , 1)

t. : 3.8(Gershgorins Disk Theorem),

|||xk|=

j akjxj j |akjxj | j |akj ||xk|= k(A)|xk| (A)|xk|

. , ||= (A), ()(=). ,

(i)

j akjxj= j |akjxj |,

(ii)

j |akjxj |=

j |akj ||xk|,

(iii) k(A) = (A)

. (i),

akjxj = cj z, (1 j n), |z|= 1

real number c1, . . . , cn 0 z C ( 3.11 ). , A> 0, (ii)

|xj | = |xk|, (1 j n)

. , A> 0,

akj |xk| = akj |xj | = |cj z| = cj , (1 j n)

. ,

xj =cjakj

z = |xk|z, (1 j n)

. , X = |xk|z(1, 1, . . . , 1)t. eigen-vector 1 = (1, 1, . . . , 1)t . 1 eigen-

vector. A1> 0, A1= 1, > 0.30 28 3.10 [3].29, 3.9, A dominant eigen-value.30(iii).

14

3.11 1, . . . , r C

|1 + + r| = |1|+ + |r|

,

i = ci, (i = 1, . . . , r)

C 0 c1, . . . , cr R . ||= 1 OK.31

3.12 A Mn,n(C) , A > 0 . A || = (A)eigen-value . = (A)> 0, dimEA = 1

.32

3.13 ,

(2 00 1

)

(0 1

1 0

) , 3.10 A

positive . .33

. :

3.14 A= (aij)Mn,n(C), A matrix norm

A = max{|aij |

1 i, j n}.34

. .

3.15 A,B Mn,n(C), .

() A = 0, A> 0.

() cC, cA= |c|A.

() A+B A+B.

() AB nAB.

31Hint: r. [I, , 7.5] .32 [I, 7.6.24()] . ( (1, 1, . . . , 1)t eigen-vector.)33 Markov matrix.34 matrix norm . max norm .

Mn,n(C) =Cn2 identify, max norm L-norm. Matrix norm exp(A)

(1.2 ).

15

4 Markov Matrix

version

150909 2 toy example .

3 Markov matrix.

4.1 AMn,n(C) Markov matrix,

() j(A) = (A) = 1 for all j = 1, . . . , n.

() C A eigen-vector, || 1.() 1 is a dominant eigen-value of A.

: () Markov matrix . () 3.9

(Gershgorins Disk Theorem ) () . () 2.3

().

positive Markov matrix.

4.2 AMn,n(C) positive Markov matrix,

() C A eigen-vector , = 1, ||< 1. , 1 Aunique dominant eigen-value.

() dimEA1 = 1.

: 3.12( 3.10 transpose version) 4.1 direct con-

sequence.

4.3 A Mn,n(C) right Markov matrix , 4.1 4.2.

4.2, A positive Markov, dimEA1 = 1

, dimEA1 = multA(1) .

Markov chain essence !

easy exercise. ( 1= (1, 1, . . . , 1)t.)

4.4 .

() 0AMn,n(C) Markov matrix At 1= 1.() 0X Cn probability vector Xt 1= (1)11.

16

4.5 .

() A,B Mn,n(C) Markov matrix, AB Markov matrix.

() A Mn,n(C) Markov matrix, X Cn probability vector,AX probability vector.

4.6 A Mn,n(C) right Markov matrix, 4.4 4.5.35

Final touch matrix norm.36

4.7 AMn,n(C) (right) Markov matrix, dimEA1 = multA(1)(,A 1-Jordan block ).

: (i) m , Am Markov( 4.4()

), Am 1.

(ii) U1AU = J A Jordan canonical form,

Jm = U1AmU n2U1Am|U n2U1U

. ,{Jm

m 0} bounded above.(iii) J 1-Jordan -block (1 1)- ( 3.2 ). , J 1-Jordan block( 3.6 ).

(iv) A right Markov [ I, 7.6.24] .

article main theorem state.

4.8 AMn,n(C) positive (right) Markov matrix,

() dimEA1 = multA(1) = 1.37 ( A unique dominant eigen-value 1

dominant eigen-vector (up to scalar) unique.)

() limm

Am.

: () 4.1, 4.2 4.7.

() 3.5. 35X probability vector, AX probability vector.36, 4.7 4.1, first touch OK.37 multA(1)= 1, dimE

A1 =1. ?

17

4.9 A Mn,n(C) positive Markov matrix , limm

Am =L

. A unique dominant eigen-value 1 dominant

eigen-vector unique probability vector P ,

() AL=LA=L.

() L Markov matrix, L= (P, P, . . . , P ).

() probability vector X , limm

AmX =LX = P .38

() P > 0. ( L> 0.)

: () AL=A limm

Am = limm

Am+1 =L. (LA=L.)

() Am Markov( 4.5),

1t L = 1t limm

Am = limm

1t Am = limm

1t = 1t

( L 0), L Markov( 4.4). , L columnprobability vector. , AL = L, L column eigen-

value 1 A eigen-vector. , L column P .

() P = (p1, . . . , pn)t, X = (x1, . . . , xn)

t . , L = (P, P, . . . , P )

, LX i-

pix1+ +pixn = pi(x1+ +xn) = pi

. , LX = P .

() . AP =P , A> 0, P 0, P > 0.

, positive Markov matrix A , probability

vector X AX, A(AX), A(A2X), . . . limm

AmX

, A dominant eigen-vector P .

( limm

Am = L. ?)

power method(1.8), , X depend

. , P .

4.9 dominant eigen-vector P Perron-Frobenius

vector, stochastic vector, stationary vector, fixed probability vector

.39

38, , LP =P .39 6 PageRank vector.

18

.

4.10 Ai Mn,n(C) positive Markov matrix , block diagonalmatrix A= diag(A1, . . . , Ak).

() limm

Am.

() Eigen-space EA1 basis.

Right Markov matrix. (.)

4.11 A Mn,n(C) positive right Markov , limm

Am = L

. At unique dominant eigen-value 1 dominant

eigen-vector unique probability vector P ,40

() AL=LA=L.

() L right Markov matrix, L= (P, P, . . . , P )t.

() probability vector X , Xt L=P t,41 LX = (P t X) 1.

comment().

4.12 () , A diagonalizable

, article . Jordan

canonical form.

() A Markov matrix , As > 0 s , A

regular Markov matrix( primitive Markov matrix) . Regular

Markov matrix 4.2 4.8 4.9

4.11 .42 [ ] : A eigen-value

||= 1, s As > 0 eigen-value, s =1( 4.2). , As+1 > 0, s+1 = 1. = 1. dimEA

s

1 = 1

( 4.8), EA1 =EAs

1 .

, article Markov chain( Markov process)

chain .

, . .43

40A dominant eigen-vector 1( 2.3).41 Right Markov right right notation( ). , L

act.42, n, A regular.43Wikipedia . stochastic process.

19

5 Hardy-Weinberg Equilibrium

version

180102 Population Genetics ( ) .44

19081909 Hardy-Weinberg equilibrium(principle, law)

.45

5.A.

, .

allele() (T, t) . genotype TT Tt

phenotype, genotype tt phenotype.46

, m 0, m-(m-) genotype frequency

pm = Probm-(TT ), qm = Probm-(Tt), rm = Probm-(tt)

, m- genotype frequency vector Pm = (pm, qm, rm)t

.47 (0-) genotype frequency vector

P0 = (p0, q0, r0)t= (p, q, r)

t

. m- allele frequency

am = Probm-(T ), bm = Probm-(t)

, m- allele frequency vector Qm= (am, bm)t .

0- allele frequency vector

Q0 = (a0, b0)t= (a, b)

t

. , Pm, Qm 0,

pm+ qm+ rm = 1, am+ bm = 1

.

44 population , . . .

45G. H. Hardy(18771947) Hardys Theorem, Hardys Inequality, Hardy-Littlewood

Theorem (). 20 neo-Darwinism(modern

synthesis) Hardy. (W. Weinberg(18621937) ().)46Mendel. Dominant trait recessive trait .47Frequency vector = probability vector.

20

, m- allele frequency

am = pm+12 qm, bm =

12 qm+ rm

( ?).48 0-

a = p+ 12 q, b =12 q+ r

.

, 1- genotype frequency. TT -type

genotype table(Markov matrix)

ATT = [ TT -type]

TT Tt tt

TT 1 12 0

Tt 0 12 1

tt 0 0 0

. Tt-type tt-type

genotype Markov matrix

ATt = [ Tt-type] Att = [ tt-type]

TT Tt tt

TT 1214 0

Tt 1212

12

tt 0 1412

TT Tt tt

TT 0 0 0

Tt 1 12 0

tt 0 12 1

. ,

A = pATT + qATt+ rAtt

,

A =

p+ 12 q

12 p+

14 q 0

12 q+ r

12 p+

12 q+

12 r p+

12 q

0 14 q+12 r

12 q+ r

=a 12 a 0

b 12 a

0 12 b b

. A Markov matrix.

48, , allele frequency genotype frequency.

21

Markov matrix A 2 toy example,

A =

a 12 a 0

b 12 a

0 12 b b

=TT TT Tt TT tt TTTT Tt T t Tt tt TtTT tt T t tt tt tt

. , , TT Tt () TT() Tt . , 2 toy example

,49 1- genotype frequency vector

P1 =

p1

q1

r1

= AP0 =a 12 a 0

b 12 a

0 12 b b

p

q

r

=

ap+ 12 aq

bp+ 12 q+ar12 bq+ br

=

a2

2ab

b2

().50 , 1- allele frequency vector

Q1 =

(a1

b1

)=

(a2+ 12 2ab12 2ab+ b

2

)=

(a

b

)= Q0

. , allele frequency !

, 2- genotype frequency vector

P2 =

p2

q2

r2

=a1

12 a1 0

b112 a1

0 12 b1 b1

p1

q1

r1

=a 12 a 0

b 12 a

0 12 b b

a2

2ab

b2

=

a3+a2b

a2b+ab+ab2

ab2+ b3

=

a2

2ab

b2

= P1, allele frequency vector Q2 = (a, b)

t=Q0. ,

Pm =

pm

qm

rm

= AmP0 =

a2

2ab

b2

= P1, Qm =(a

b

)= Q0, (m 1)

.

49= .50 , , mathematical model random mating, sex

independent genotype frequency . (

) . , (, m- m-).

22

,51 allele frequency vector (a, b)t constant

, genotype frequency vector equilibrium(Hardy-

Weinberg equilibrium) (a2, 2ab, b2)t . (20 neo-Darwinism

(modern synthesis).)

equilibrium , ,

A regular Markov matrix.52 , P1 = (a2, 2ab, b2)t

A dominant eigen-vector.53 , limm

AmP0 = P1.

5.B. Sex Linked Gene

, () X- .

allele (X,x). genotype

XX, Xx, xx xx, genotype XY, xY xY

.54

, m- genotype frequency

pm = Probm-(XX), qm = Probm-(Xx), rm = Probm-(xx)

, m- genotype frequency vector Pm = (pm, qm, rm)t

. ( Pm 0, pm + qm + rm = 1.) m- allelefrequency

am = Probm- (X), bm = Probm- (x)

( am, bm 0, am+ bm =1), 5.A,

am = pm+12 qm, bm =

12 qm+ rm, (m 0)

. m- genotype frequency

cm = Probm-(XY ) = Probm- (X),

dm = Probm-(xY ) = Probm- (x)

( allele frequency genotype frequency ), m-

allele frequency vector Rm = (cm, dm)t.

51 (!) [ : = 3 : 1]. ?52a, b =0, A2> 0. 4.12().53A eigen-value 1, 1

2, 0. , , A diagonalizable, det(A) = 0.

54.

23

, (m+1)- genotype frequency. 5.A, genotype Markov matrix

AXY = [ XY -type] AxY = [ xY -type]

XX Xx xx

XX 1 12 0

Xx 0 12 1

xx 0 0 0

XX Xx xx

XX 0 0 0

Xx 1 12 0

xx 0 12 1

. ,

Am = cmAXY + dmAxY =

cm

12 cm 0

dm12 cm

0 12 dm dm

, (m 0)(Am Markov matrix), (m+1)- genotype frequency

vector

Pm+1 =

pm+1

qm+1

rm+1

= AmPm =cm

12 cm 0

dm12 cm

0 12 dm dm

pm

qm

rm

=

amcm

dmpm+12 qm+ cmrm

bmdm

=

amcm

1amcm bmdmbmdm

.55

5.1 () (m+1)- genotype(allele) frequency

table(Markov matrix) BXX , BXx, Bxx

.

() Bm = pmBXX + qmBXx+ rmBxx,

Rm+1 =

(cm+1

dm+1

)= BmRm =

(am am

bm bm

)(cm

dm

)=

(am

bm

)

.

55 vector probability vector( 4.5), .

24

() , , genotype

( ?), . , dm+1 = bm. ,

allele frequency bm+1

bm+1 =12 (1amcm bmdm)+ bmdm

= 12(1 (1 bm)(1dm) bmdm

)+ bmdm

= 12 (bm+ dm)

. M =

(12

12

1 0

), Xm =

(bm

dm

),

Xm+1 =

(bm+1

dm+1

)=

(12

12

1 0

)(bm

dm

)= MXm

. ,

X1 = MX0, X2 = MX1 = M2X0, . . . , Xm = M

mX0

.

, M2 > 0, M regular right Markov matrix.

, 4.11 ( 4.12() ). M t

dominant eigen-vector(23 ,

13

)t(),

limm

(bm

dm

)= lim

mMm

(b0

d0

)=

(23

13

23

13

)(b0

d0

)=

(23 b0+

13 d0

23 b0+

13 d0

)

. , , allele frequency vector

allele frequency vector . , ,

allele frequency vector,

b = limm

bm =23 b0+

13 d0, a = limm

am = 1 b

, Hardy-Weinberg equilibrium

limm

pm

qm

rm

= limmam1cm1

bm1dm1

= limmam1am2

bm1bm2

=

(a)2

2ab

(b)2

( 5.1).56

56.

25

, , 112 = 0.0833

, 1200 = 0.0050 . ,

limm

dm = b,

limm

rm = (b)2.

(112

)2= 0.0069,

() .57

5.C. Generalization

. :

() r- T, S, . . . ( genotype

3r-),

() Allele pair (T, t), s-tuple (T1, . . . , Ts)( genotype

(s+12

)-),

() ,

() ()() genotype frequency,

() ()() sex linked gene,

() mutation, sexual selection,

mathematical model

.

(. (?).)

5.2 20 neo-Darwinism(modern synthe-

sis) G. H. Hardy Hardy ,

, . , Hardy I have never done anything

useful. No discovery of mine has made, or is likely to make the least differ-

ence to the amenity of the world ,

. Hardy cricket team geneticist

, () . Hardy thus became the

somewhat unwitting founder of a branch of applied mathematics.58

57 (0.019) . . ( limm

rm = (b)2

initial condition( founder effect).)58Wikipedia.

26

6 Google Ranking System

version

180102 Markov chain PageRank algorithm(Google ranking

system) . Internet search engine (

web page ). ranking system

: search result(rank) ?

1995(?) Stanford University Computer Science

L. Page S. Brin G. H. Golub[4] Matrix Computation

(Golub dominant eigen-vector power method). Page Brin

,59 dominant eigen-

vector Marcov chain.

PageRank idea :

. ,

.

6.1 i ri(, 0 ri R) ? follower , i

. , follower , ri

j i rj

. j i j i.,

ri =j i

1

Njrj , , Nj =

{k | j k}.60

, ? ?

(Golub (?)) ? .

59 (1998) [6] eigen-value

. [6] T. Winograd Page .Google ,

honor system . [6]

, , .60, 0 follower 0.

27

6.1, rank(, reputation, , ) ri

. , rank.

800 !61

6.2 web page i rank ri

ri =j i

1

Njrj , , Nj =

{k | j k} (, 0 ri R).62 j i web page j webpage i link.

, ri ? ri ?

ri 1- ? ,

aij =

1

Nj(if j i and i = j)

0 (otherwise)

, 1-

ri =

j aij rj

. ( aii = 0. , self-link.) , , A= (aij)Mn,n(R), rank vector R= (rj)Rn,

AR = R

. , R eigen-value 1 A eigen-vector. ,

A matrix size n 230.

6.3 A zero column, A Markov matrix

. (, A dominant eigen-value 1.)

, A zero column , outlink web

page A eigen-value 1 . (

R eigen-value 1 , .)

61 2017 Google 800.62 web page.

28

R , R (up to scalar multiple) ? , dimEA1

1 ? multA(1) ? A 0 A very very sparse matrix.63 , 0 .

, A (after renumbering) block diagonal matrix

. (Page-Brin[6] there is a small problem , two web pagesthat point to each other but to no other page . , A

diagonal block

(0 1

1 0

) . , (1)

A eigen-value, 1 unique dominant eigen-value.) ,

multA(1) diagonal block (

4.10).64

dominant eigen-vector Marcov chain

, .

, 1.5 , A 230

.65

1- : A Mn,n(C) zero column , zerocolumn 1n 1 (, 1 = (1, 1, . . . , 1)

t). A

, A Markov matrix( 6.3 ).

, outlink page(

) page outlink . ,

( ) (?).

1n , A A . (

.)

, Markov matrix A very very sparse.

6.2 , .

63Sparse matrix non-zero entry.64 diagonal block ,

, .65 PageRank algorithm , 1.5, 230

.

29

Page-Brin 800 () :

theory positive Markov matrix, A

positive Markov matrix ! ,

A 230 .

2- : Google matrix GMn,n(R)

G = dA+1dn

1n

(, 1n Mn,n(C) 1).

d = 0.85

(d damping factor). G positive Markov matrix

().

Google matrix G = (gij) . Nj = 0

(, j -page outlink , A j -th column 0,

A j -th column 1n 1), G j -th column1n 1. ,

Nj = 0, gij =1n for all i. , Nj = 0

gij =

0.15

n+

0.85

Nj(if j i and i = j)

0.15

n(otherwise)

. (, G positive Markov.)

, 1n0.15n

, 1Nj0.85Nj 0.15Nj . ,

A non-zero component 1Nj . ,

damping factor, , 0.5 0.88, 0.85

? damping factor 0.9999 1 1230 , 6.2 ?66

, damp,

/ . , limm

GmX

(, X Rn probability vector).66 0 < d < 1, G positive Markov matrix.

30

G positive Markov matrix, Perron-Frobenius vector

limm

GmX = P theory( 4.8 4.9)

(, X Rn probability vector). dominant eigen-vector P PageRank vector (P i- i-page

PageRank).67 P power method. Page-Brin[6]

, G52X.68 69

damping factor d = 0.85 Page-Brin[6]

.70 Random Surfer Model.

Random Surfer Model G (i, j)- gij j -page random

surfer i-page ( ) . , X web

page ,71 - lim

mGmX . ( 2 toy example .) ,

backlink (, ) page surfer

. , (? !) , page surfer

page outlink click 85%, outlink

15% .72 (, j i , j -page surfer i-page outlink click i-page

0.15n , (j -page)0.15n .)

PageRank algorithm implement,

() .73 , ()

, .

676.2 rank vector.68 [2], can be computed in a few hours on a medium size workstation.69, X PageRank vector, .70 [6] 0.85, [2].71 (random) surfer.72Random surfer 85% randomly . ,

random imaginary virtual.73 0.85().

31

7 Perron-Frobenius Theory

version

160102 Perron-Frobenius Theory

. 4 ,

4 . ,

, article

. Meyer[5].

A> 0. A 0, .74

7.1 C,D Mm,n(C) , .

() C = (cij), |C| =(|cij |

)Mm,n(C) .75 X Cn

, |X|.() C >D if and only if CD> 0.() C D if and only if CD 0.

main theorem, .

7.2 (Perron-Frobenius Theorem : Positive Case) A Mn,n(C)A> 0 , A dominant eigen-value . :

() = ||> 0.() AX = X( 0 =X Cn), A |X|= |X|, |X|> 0.() dimEA = multA() = 1.

, (), A> 0 unique dominant eigen-value > 0

. , (), positive eigen-vector

. , () , positive eigen-vector

probability vector P , P EA = P . P A Perron-Frobenius vector.

7.3 A Mn,n(C) , m(A) = max{|| A eigen-value}

. (, A dominant eigen-value, ||=m(A).)

74 A > 0 Perron Theory , A 0 Perron-FrobeniusTheory. Meyer[5]

75, |C| det(C).

32

AMn,n(C), A> 0.

7.4 .

() m(A)> 0.

() 1m(A) A> 0, m(

1m(A) A

)=1.

7.5 0 = cC, .

() EA =EcAc .

() A(t) = (t1) (tn), cA(t) = (t c1) (t cn).76 ,multA() = multcA(c).

7.2 , , 3.10 4.7 modify

. technical .

7.6 () m(A) A eigen-value.

() AX = X(, ||=m(A), 0 =X Cn), A |X|=m(A) |X|, |X|> 0.

: (i) (notational convenience) A 1m(A) A normalize,

m(A) = 1().

(ii) ||=m(A) = 1, AX = X(, 0 =X Cn). ,

|X| = || |X| = |X| = |AX| |A| |X| = A |X|

. A |X|= |X|, A |X| = |X|, A

(A|X||X|

)> 0( ?). , Z =A|X|

, Z > 0. A(Z|X|

)> Z > 0.

AZZ > Z, 11+AZ >Z, B =1

1+A

, BZ >Z. ,

B2Z = B(BZ) > BZ > Z, . . . , BmZ > Z, (m 1)

. , m(A) = 1, m(B) = 11+ < 1, limmBm = 0

( 3.2 3.5 ). , BmZ >Z limit

, 0 Z. ! , A |X|= |X|=m(A) |X|. , A |X|> 0, A |X|= |X|> 0.

76Hint: .

33

7.7 A dominant eigen-value(, ||=m(A)),

() = ||> 0.

() dimEA = multA() = 1.

: (i) A normalize, m(A) = 1 = || ().

(ii) 0 =X = (x1, . . . , xn)t Cn A eigev-vector(, AX = X). , 7.6 , A |X| = |X|> 0 . ,

A |X| = |X| = || |X| = |X| = |AX|

, i () (, 1 i n), i-,j aij |xj | =

j aijxj

. , 3.11,

aijxj = cj z, , xj = zcjaij

(1 j n)

c1, . . . , cn 0 z C . eigen-vector Y =

(c1ai1

, . . . , cnain

)t.77 ( dimEA =1

. Y X depend.78)

(iii) (ii) eigen-value A eigev-vector Y 0 . , , EA vector 0

0 . , dimEA = 1

EA probability vector

( ?). P,Q EA probability vector .

0 = P Q EA , P Q 0, P Q . . dimEA =1.

(iv) (ii), |X|> 0, c1, . . . , cn = 0, Y > 0. ,

Y = AY = |AY | = |Y | = || Y = Y

, = 1= ||.

77 i.78 gap.

34

(v) dimEA = multA() , dimEA = multA()

. J = U1AU A Jordan canonical form

, limm

Jm=( 3.2 3.14 ). ,

Jm = U1AmU n2U1UAm

( 3.15()), limm

Am=. Am =(a(m)ij

)

, Am = a(m)imjm . , Y = (y1, . . . , yn)t

, AY = Y ,

yim =

j a(m)imj

yj (

j a(m)imj

)min

k{yk} Ammin

k{yk}

. .

7.2 ( 7.6 7.7) . , A > 0

, A positive dominant eigen-vector probability vector P

, P . P A Perron-Frobenius vector. ,

At > 0, At Perron-Frobenius vector Q(At

dominant eigen-value ).

Perron-Frobenius Theorem.

7.8 X 0 A eigen-value eigen-vector,=m(A). ( X Perron-Frobenius vector P positive scalar multiple.)

: Q At Perron-Frobenius vector. ,

At Q=m(A)Q, Qt X > 0. , AX = X,

Qt X = Qt (X) = (Qt A)X = m(A)Qt X

, =m(A).

7.9 (Collatz-Wielandt Formula) N = {X Cn |X 0 and X = 0}, f : N R

f(X) = f((x1, . . . , xn)

t)= min

{AX i-

xi

1 i n, xi =0}, (X N ), m(A) = max{f(X) |X N}.

35

version

180102 [1] M. Artin, Algebra, Prentice-Hall, 1991.

[2] S. Brin and L. Page, The anatomy of a large-scale hypertextual Web search

engine, Computer Networks and ISDN Systems, 30, 107117, 1998.

[3] S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed.,

Pearson, 2002.

[4] G. H. Golub and C. F. Van Loan, Matrix Computation, 4th ed., JHU

Press, 2012.

[5] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

[6] L. Page, S. Brin, R. Motwani and T. Winograd, The PageRank citation

ranking : bringing order to the web, Technical Report, Stanford InfoLab,

1999.

36