D.4 First Order Linear Systems
Here is a discussion of linear systems of the form
where A is a constant n×n matrix and f is a vector valued function having all entries continuous.
Of course the existence theory is a very special case of the general considerations above but I will
give a self contained presentation based on elementary first order scalar differential equations and
Definition D.4.1 Suppose t → M
is a matrix valued function of t. Thus M
In words, the derivative of M
is the matrix whose entries consist of the derivatives of the
entries of M
. Integrals of matrices are defined the same way. Thus
In words, the integral of M
is the matrix obtained by replacing each entry of M
integral of that entry.
With this definition, it is easy to prove the following theorem.
Theorem D.4.2 Suppose M
are matrices for which M
makes sense. Then
both exist, it follows that
In the study of differential equations, one of the most important theorems is Gronwall’s
inequality which is next.
Theorem D.4.3 Suppose u
0 and for all t ∈
where K is some nonnegative constant. Then
Proof: Let w
Then using the fundamental theorem of calculus, 4.10 w
satisfies the following.
Multiply both sides of this inequality by e−Kt and using the product rule and the chain
Integrating this from 0 to t,
Now multiply through by eKt to obtain
Therefore, 4.12 implies
With Gronwall’s inequality, here is a theorem on uniqueness of solutions to the initial value
in which A is an n × n matrix and f is a continuous function having values in ℂn.
Theorem D.4.4 Suppose x and y satisfy 4.13. Then x
for all t.
Proof: Let z
. Then for
Note that for K = max
(For x and y real numbers, xy ≤
because this is equivalent to saying
Now multiplying 4.14 by z and observing that
it follows from 4.15 and the observation that z
and so by Gronwall’s inequality,
= 0 for all t ≥
for all t ≥ a.
Now let w
0. Then w′
and you can
repeat the argument which was just given to conclude that
t ≤ a. ■
Definition D.4.5 Let A be an n × n matrix. We say Φ
is a fundamental matrix for A
−1 exists for all t ∈ ℝ.
Why should anyone care about a fundamental matrix? The reason is that such a matrix valued
function makes possible a convenient description of the solution of the initial value
on the interval,
First consider the special case where n
This is the first order linear
where g is a continuous scalar valued function. First consider the case where g = 0.
Lemma D.4.6 There exists a unique solution to the initial value problem,
and the solution for λ = a + ib is given by
This solution to the initial value problem is denoted as eλt. (If λ is real, eλt as defined here reduces
to the usual exponential function so there is no contradiction between this and earlier notation
seen in calculus.)
Proof: From the uniqueness theorem presented above, Theorem D.4.4, applied to the case
where n = 1, there can be no more than one solution to the initial value problem, 4.19. Therefore,
it only remains to verify 4.20 is a solution to 4.19. However, this is an easy calculus exercise.
Note the differential equation in 4.19 says
With this lemma, it becomes possible to easily solve the case in which g≠0.
Theorem D.4.7 There exists a unique solution to 4.18 and this solution is given by the
Proof: By the uniqueness theorem, Theorem D.4.4, there is no more than one solution. It
only remains to verify that 4.22 is a solution. But r
and so the
initial condition is satisfied. Next differentiate this expression to verify the differential
equation is also satisfied. Using 4.21
, the product rule and the fundamental theorem of
Now consider the question of finding a fundamental matrix for A. When this is done, it will be
easy to give a formula for the general solution to 4.17 known as the variation of constants formula,
arguably the most important result in differential equations.
The next theorem gives a formula for the fundamental matrix 4.16. It is known as Putzer’s
Theorem D.4.8 Let A be an n×n matrix whose eigenvalues are
to multiplicity as roots of the characteristic equation. Define
and let the scalar valued functions, rk
be defined as the solutions to the following initial value
Note the system amounts to a list of single first order linear differential equations. Now
Furthermore, if Φ
is a solution to 4.23 for all t, then it follows
−1 exists for all t and
is the unique fundamental matrix for A.
Proof: The first part of this follows from a computation. First note that by the
Cayley Hamilton theorem, Pn
= 0 and
= 0. Also from the formula, if we define
to correspond to the above definition, for all k ≥
Now for the computation:
It remains to verify that if 4.23 holds, then Φ
exists for all t.
To do so, consider v≠0
suppose for some t0,
But also z
and so by the theorem on uniqueness, it must be the case that z
for all t,
and in particular for t
and so v = 0, a contradiction. It follows that Φ
must be one to one for all
and so, Φ
exists for all t.
It only remains to verify the solution to 4.23 is unique. Suppose Ψ is another fundamental
matrix solving 4.23. Then letting v be an arbitrary vector,
both solve the initial value problem,
and so by the uniqueness theorem, z
showing that Φ
for all t.
is arbitrary, this shows that Φ
It is useful to consider the differential equations for the rk for k ≥ 1. As noted above, r0
Sometimes people define a fundamental matrix to be a matrix Φ
such that Φ
0 for all t.
Thus this avoids the initial condition, Φ
The next proposition
has to do with this situation.
Proposition D.4.9 Suppose A is an n×n matrix and suppose Φ
is an n×n matrix for each
t ∈ ℝ with the property that
Then either Φ
−1 exists for all t ∈ ℝ or
−1 fails to exist for all t ∈ ℝ.
Proof: Suppose Φ
exists and 4.24
holds. Let Ψ
so by Theorem D.4.8, Ψ
exists for all t.
also exists for all t.
Next suppose Φ
does not exist. I need to show Φ
does not exist for any
Suppose then that Φ
does exist. Then letΨ
Ψ so by Theorem D.4.8
it follows Ψ
exists for all t
and so for all
must also exist, even for t
which implies Φ
exists after all.
The conclusion of this proposition is usually referred to as the Wronskian alternative and
another way to say it is that if 4.24 holds, then either det
= 0 for all
never equal to 0.
The Wronskian is the usual name of the function, t →
The following theorem gives the variation of constants formula,.
Theorem D.4.10 Let f be continuous on
and let A be an n×n matrix and x0 a vector in
ℂn. Then there exists a unique solution to 4.17, x, given by the variation of constants
the fundamental matrix for A. Also,
t,s and the above variation of constants formula can also be written as