Principles of Mathematics in Operations Research

60 4 Eigen Values and Vectors

v 2 = (l,2,4)T, B 2 =. Therefore,

B =

where Ai

10

01

-12

B-

0 2-1

-2 5-2

1 -2 1

=>r'AB

01

-12

01

-1 2

and A 2 = [2].

4.4 Powers of A

Example 4.4.1 (Compound Interest) Let us take an example from engi-

neering economy. Suppose you invest $ 500 for six years at 4 % in Citibank.

Then,

Pk+i = 1.04Pfc, P 6 = (1.04)^6 , P 0 = (1.04)^6 500 = $632.66.

Suppose, the time bucket is reduced to a month:

0.04

~12~

Pk+i= (l + ^W Pr2= (l

What if we compound the interest daily?

0.04

72

Po = (1.003)^72 500 = $635.37.

Pk+l = ( 1 + ^ J Pk, i>6(364) + 1.5 =(^1 + 36^ J

2185.5

P 0 = $635.72.

Thus, we have

Pk+i - Pk

At

= 0.04Pfc -»

dP

dt

0.04P ^> P(t) = eomtP 0.

In the above simplest case, what we have is a difference/differential equa-
tion with one scalar variable. What if we have a matrix representing a set of
difference/differential equation systems? What is e~Ai 1

Example 4.4.2 (Fibonacci Sequence)

Fk+2 = Fk+i + Fk, Fi = 0, F 2 — 1.

Uk =

Fk+i

Fk

Uk+l —

Fk+2

Fk+i

1 1

10

Fk+1

Fk

• Auk.

uk = Aku 0

Hence, we sometimes need powers of a matrix!

1 1

10