Principles of Mathematics in Operations Research

62 4 Eigen Values and Vectors

Ai = 1.0, A 2 = 0.94 =>• V! , V 2

SAS'^1 =

|1

-1 1

1.00

Vk

Zk

0.95 0.01

0.05 0.99

-ik

|1

-1 1

0.94

1.00*

-1

5 5

5 _I

6 6

0.94*

/5 5

= ^2/o + ^o

Since 0.94* -» 0 as A; -)• oo,

+ I |»b " ^o ) 0.94*

I I

I -I

1

-1

2/oo

•2 CO

= l^2/o + |«b L6 6 J

TVie steady-state probabilities are computed as in the classical way, Auoo —

1 • Moo, corresponding to the eigen value of one. Thus, the steady-state vector

is the eigen vector of A corresponding to A = 1, after normalization to have

legitimate probabilities (see Remark 4-3-4):

Moo = avi = -^5 [I

6

" 1 '

5

L J^1

" 1 "

6

5

6

4.4.2 Differential Equations

Example 4.4.7

du

~dl

Au •

23

14

u <=> u(t) = eAtu 0 -

Xx = 5,vi = (l,lf, A 2 = l, «2 = (-3,l)a

u(t) = a 1 eAlS 1 + a 2 e*2tv 2 = axe „5t

U 0 = «1 + a 2

u(t) 1 -3

1 1

a5t

-3

1

OLi

1

1

1 -3

1 1

+ a 2 et

Oil

-3

1

= S

J>t

5-^0-

The power series expansion of the exponentiation of one scalar is