TABLE OF CONTENTS Up to this point, almost everything was either motivated by or evolved from the consideration of systems of linear algebraic equations. But we have come to a turning point, and from now on the emphasis will be different. Rather than being concerned with systems of algebraic equations, many topics will be motivated or driven by applications involving systems of linear differential equations and their discrete counterparts, difference equations.
For example, consider the problem of solving the system of two first-order linear differential equations, du 1 /dt = 7 u 1 ? 4 u 2 and du 2 /dt = 5 u 1 ? 2 u 2 . In matrix notation, this system is
| (7.1.1) |  |
where 
. Because solutions of a single equation u ? = ?u have the form u = ?e ?t, we are motivated to seek solutions of (7.1.1) that also have the form
| (7.1.2) |  |
Differentiating these two expressions and substituting the results in (7.1.1) yields
In other words, solutions of (7.1.1) having the form (7.1.2) can be constructed provided solutions for ? and 
in the matrix equation Ax = ? x can be found. Clearly, x = 0 trivially satisfies Ax = ? x, but x = 0 provides no useful information concerning the solution of (7.1.1). What we really need are scalars ? and nonzero
