Principles of Mathematics in Operations Research

4.4 Powers of A

4.4.1 Difference equations

Theorem 4.4.3 If A can be diagonalized (A — SAS"^1 ), then

uk = Aku 0 = (SAS-^iSAS-^1 ) • • • {SAS-^uo = SAkS~^1 u 0 -

Remark 4.4.4

61

Uk [vi,-

x

k

A_

S^1 u 0 = aiXiVi H hanX„vn.

The general solution is a combination of special solutions XkVi and the coeffi-

cients a, that match the initial condition UQ are aiXlvi + • • • + a„A°un = Uo

or Sa — uo or a = 5 _1MO- Thus, we have three different forms to the same

equation.

Example 4.4.5 (Fibonacci Sequence, continued)

1 1

10

A, =

d(s) =

l + \/5

s-1 -1

-1 s

A 2 =

Ai A2

1 1

A = SAS-^1

uk = AkuQ -

Ax

A 2

= s* - s - 1 = 0.

1-\/5

1

2 '

1 -A 2

-1 x 1

•ffc+i

Fk

Fk =

Xk A 2

Ai — A2

k

X\ — A2

Ai A2

1 1

1

71

x

k

At

1 + v^

1

Ai — A2

1

-1 M A 2

'1-VET

1000

Since -4? (^—f^-J <\, Fww = the nearest integer to 4= (^fl-)

Note that the ratio -J^^1 = 1-y^5 = 1.618 is known as the Golden Ratio, which

represents the ratio of the lengths of the sides of the most elegant rectangle.

Example 4.4.6 (Markov Process) Assume that the number of people leav-
ing Istanbul annually is 5 % of its population, and the number of people en-
tering is 1 % of Turkey's population outside Istanbul. Then,

#inside

jfcoutside

0.95 0.01

0.05 0.99

A

0.95 0.01

0.05 0.99

d(s)

s-0.95 -0.01

-0.05 s-0.99

= (s-1.0)(s-0.94).