Page 314 - A Course in Linear Algebra with Applications

298 Chapter Eight: Eigenvectors and Eigenvalues

2
2
because of the identity cosh (kt) — sinh (A;i) = 1. Therefore
at the time when Division B has lost all of its tanks, Division
A still has a tanks where a 2 — 0 = a§ — &o- Hence the number
of tanks that Division A has left at the end of the battle is

a
b
V o - o-
Not surprisingly, since it had more tanks to start with, Divi-
sion A wins the battle.
However, there is a way in which Division B could con-
ceivably win. Suppose that

— a 0 < b Q < a 0.
v 2
Suppose further that Division A consists of two columns with
equal numbers of tanks, and that Division B manages to at-
tack one column of Division A before the other column can
come to its aid. Since 6 0 > a 0 , Division B defeats the first
|
2
column of Division A, and it still has •Jb ) — \OQ tanks left.
Then Division B attacks the second column and wins with

b
a
b
a
y l - i l - 4°o = y l - 2 o
tanks left.
Thus Division B wins the battle despite having fewer
tanks than Division A: but it must have more than ao/^2
or 71% of the strength of the larger division for the plan to
work. This explains the frequent success of the "divide and
conquer strategy".
Higher order equations
Systems of linear differential equations of order 2 or more
can be converted to first order systems by introducing addi-
tional functions. Once again the procedure is similar to that
adopted for systems of linear recurrences.