Page 302 - A Course in Linear Algebra with Applications

286 Chapter Eight: Eigenvalues and Eigenvectors

are returned, while 10% remain lent out and 10% are reported
lost. Finally, 25% of books listed as lost the previous month
are found and returned to the library. How many books will
be in the library, lent out, and lost in the long run?
Here there are three states that a book may be in: Si =
in the library: S2 = lent out: S3 = lost. The transition matrix
for this Markov process is

.8 .8 .25'
P= I .2 .1 0
0 .1 .75

Clearly P 2 has positive entries, so P is regular. Of course P
has the eigenvalue 1; the corresponding eigenvector with entry
sum equal to 1 is found to be

So the probabilities that a book is in states Si, S2, S3 after a
long period of time are 45/59, 10/59, 4/59 respectively. There-
fore the expected numbers of books in the library, lent out,
and lost, in the long run, are obtained by multiplying these
probabilities by the total number of books, 10,000. These
numbers are therefore 7627, 1695, 678 respectively.

Exercises 8.2
1. Solve the following systems of linear recurrences with the
specified initial conditions:

l X n
(a) \ Vn+1 Z v , l z where y 0 = 0,z 0 = 1;
y +l
l
(b) \ ; : %- X f where y = <>•*> =-
z
Zn+l — Vn -r 3Z n