First of all, consider the problem
 (24.1) 
where T is an upper triangular matrix having constant coefficients and f is a continuous function of t.
Proposition 24.1.1 There exists a unique solution for t ∈ ℝ to the initial value problem 24.1.
Proof: Note that

and so the bottom row yields the initial value problem for x_{n}

We know from Section 23.1 all about solving such first order scalar equations and so there exists a unique x_{n}

Therefore, from Section 23.1, there is a unique x_{n−1}
Now with this Proposition, it is easy to verify a general existence theorem for first order systems of differential equations.
Theorem 24.1.2 Let A be an n × n matrix, let f be a continuous function of t ∈
 (24.2) 
Proof: By Schur’s theorem, Theorem 7.4.4, there is a unitary matrix U such that U^{∗}AU = T where T is upper triangular. Thus from the equation to be solved,

Let y = U^{∗}x. Then x is a solution to 24.2 if and only if y is a solution to
 (24.3) 
From Proposition 24.1.1, there exists a unique solution to ?? valid on ℝ. Thus you simply let x
Note that letting the initial condition take place at an arbitrary c is of very little interest. You can always reformulate the problem to have c = 0. If you are looking for a solution to the initial value problem where the initial condition is given at c, you could simply consider x
Theorem 24.1.4 Let A be an n × n matrix. Then it has a unique fundamental matrix Φ
Proof: Let e_{i} ≡

however, there exists a unique solution to this initial value problem from Theorem 24.1.2. ■
The fundamental matrix has some nice properties.
Theorem 24.1.5 The fundamental matrix Φ
Proof: Let x be a vector. Consider

Then by definition, y_{i}
Consider 2., 3. Fix s then letting the only variable be t, let y_{1}

and
If you can find the fundamental matrix, then you can also write down a solution to the initial value problem 24.2 in terms of the fundamental matrix. Later I will give some methods for finding the fundamental matrix, but for now, just assume you have it. I am going to derive the formula just as done earlier for first order linear scalar valued functions. The integrating factor in this case is just the inverse of the fundamental matrix. The problem then is to obtain the solution to

From Theorem 24.1.2, there exists a unique solution. It is only a matter of expressing it in terms of the fundamental matrix. Multiply both sides by Φ
 (24.4) 
Then from the product rule one obtains
 (24.5) 
This uses the product rule and the obvious fact that

where the integral is the obvious thing in which you simply integrate each entry of the vector. Therefore,
Theorem 24.1.6 Let A be a n × n matrix and let Φ

To be absolutely sure this works, just verify it satisfies the differential equation and initial condition. First of all x

Applying the fundamental theorem of calculus to the components of the various vectors and using the fact that Φ^{′}
Other than the very extensive topic of geometric theory, there really is nothing else to say about first order systems. I can’t understand how ODE books manage to drag this easy problem out for so long. They take a very easy problem and make it very difficult, but there really is still a problem. How do you find the fundamental matrix? I have shown it always exists, but is there a good way to find it? The answer is yes.
One way is to base everything on eigenvalues and eigenvectors and then generalized eigenvectors and ultimately base the entire theory on the Jordan form of a matrix which can’t even be found in all cases, although this is never made clear. I believe this is just about the worst thing you can do but it is the standard way of doing it. It did make sense in the 1950’s but it makes absolutely no sense to do it this way now when there is a much easier way based on Laplace transforms and computer algebra systems which make it very easy to find these inverse Laplace transforms when they can be found in closed forms.
In all of these considerations, I am speaking of finding the exact solutions. I believe a more significant way of considering ODE and their solutions is through the use of numerical methods applied directly to the initial value problem of Theorem 24.1.2.