We assume, that nxn matrix A is diagonalisable over , i.e. it has n linearily independent eigenvectors , ..., belonging to eigenvalues , ..., over (here or )

Assembling the eigenvectors into nxn matrix S

we can write

where is a nxn diagonal matrix with entries , ..., . Multiplying (34) on the right by S^-1 one obtains the matrix A in diagonally factorised form

Substituting (35) in (2) and multiplying obtained result on the left by S^-1 yields

or

Here a vector is defined as

Evidently, the system of differential equations (37) is a diagonal system, because the matrix D is diagonal. In explicit form the system (37) reads

Integrating (39), we obtain

or in matrix form

where

Substituting (38) in (41) and multiplying obtained result on the left by S one obtains the general solution of system (2) in terms of original variables x

The vector of integration constants can be determined satisfying the initial conditions (or boundary conditions (4)).

Satisfaction conditions gives

and

In most practical case (a=0), the formula (45) reduces to

General solution algorithm using linear algebra software is following:

1. find the eigenvalues and eigenvectors

2. compose matrices S and E(t)

3. compute the general solution to (1) by (43) or solution to the initial value problem by (46).

Example 1: Consider the system

find the general solution.

In matrix notation the system (47) becomes

The characteristic polynomial of matrix A

is

Therefore the eigenvalues are and . The eigenvector associated with is obtained by solving equation

Analogically

and therefore

Finally, we can compute the general solution of system (47) using formula (53)

Example 2: Solve the following initial value problem

For the system (55) the matrix

has eigenvalues , and . Associated eigenvectors are

respectively.

The general solution of system (55) is given by

From (44) the integration constants are

and the solution of the initial value problem (45)-(46) can be presented as

Exercises
1. Consider a linear system of differential equations

a) find the general solution
b) find the solution to the initial value problem determined by the initial conditions

2. Find the general solution to the linear system of differential equations.