First, write the system in vector and matrix form

.

Then, find the eigenvalues and eigenvectors of the matrix , denote the eigenpairs of **A** by

and .

**Assumption.** Assume that there are two linearly independent eigenvectors , which correspond to the eigenvalues , respectively. Then two linearly independent solution to are

, and

.

**Definition (Fundamental Matrix Solution)** The fundamental matrix solution , is formed by using the two column vectors .

(1) .

The general solution to is the linear combination

(2) .

It can be written in matrix form using the fundamental matrix solution as follows

.

**Notation.** When we introduce the notation

,

and

The fundamental matrix solution can be written as

(3) .

or

(4) .

**The initial condition **

If we desire to have the initial condition , then this produces the equation

.

The vector of constant can be solved as follows

.

The solution with the prescribed initial conditions is

.

Observe that where is the identity matrix. This leads us to make the following important definition

**Definition (****Matrix Exponential****)** If is a fundamental matrix solution to , then the matrix exponential is defined to be

.

**Notation.** This can be written as

(5) ,

or

(6) .**Fact.** For a system, the initial condition is

,

and the solution with the initial condition is

,

or

.

**Theorem (****Matrix Diagonalization****)** The eigen decomposition of a square matrix **A** is

,

which exists when **A** has a full set of eigenpairs for , and **d** is the diagonal matrix

and

is the augmented matrix whose columns are the eigenvectors of **A**.

.

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