Page 306 - A Course in Linear Algebra with Applications

2 9 0 Chapter Eight: Eigenvectors and Eigenvalues

If a set of n initial conditions is given, there is in fact a
unique solution of the system satisfying these conditions. For
an account of the theory of systems of differential equations
the reader may consult a book on differential equations such
as [15] or [16]. Here we are concerned with methods of finding
solutions, not with questions of existence and uniqueness of
solutions.
Suppose that the coefficient matrix A is diagonalizable,
1
so there is an invertible matrix S such that D = S~ AS is
diagonal, with diagonal entries d\,..., d n say. Here of course
the di are the eigenvalues of A. Define

X
U = S- Y.
Then Y — SU and Y' = SU' since S has constant entries.
Substituting for Y and Y' in the equation Y' = AY, we obtain
SU' = ASU, or

1
U' = {S~ AS)U = DU.
This is a system of linear differential equations for u\,..., u n,
the entries of U. It has the very simple form

d\Ui
u' 2 d 2u 2

The equation u\ = diUi is easy to solve since its differential
form is
d(ln Ui) = di.

Thus its general solution is u^ = Cie diX where ci is a con-
stant. The general solution of the system of linear differential
equations for u\,..., u n is therefore

ui =cie d i a ; , ... ,u n = c ne dnX .