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6.5 Fundamental Matrices and the
Exponential of a Matrix
Fundamental Matrices
Suppose that x1(t), . . . , xn(t) form a fundamental set of
solutions for the equation x' = P(t)x on some interval
α < t < β. Then the matrix

whose columns are the vectors x1(t), . . . , xn(t), is said to
be a fundamental matrix for the system. Note that a
fundamental matrix is nonsingular since its columns
are linearly independent vectors.
Example
Question: Find a fundamental matrix for the
system
 1 1
 x.
x' = 
 4 1
Answer:
Special fundamental matrix, Φ(t).

Sometimes it is convenient to make use of the
special fundamental matrix, denoted by Φ(t),
whose columns are the vectors x1(t), . . . , xn(t)
designated in Theorem 6.2.7. For initial
condition, x(t0) = x0,

In terms of Φ(t), the solution of the initial value
problem is
Example
For the system in previous Example, find the
fundamental matrix Φ such that Φ(0) = I2.
 Answer:

The Matrix Exponential Function
eAt
DEFINITION 6.5.1
Let A be an n × n constant matrix. The matrix
exponential function, denoted by eAt, is
defined to be

Example:
THEOREM 6.5.2

If A is an n × n constant matrix, then eAt = Φ(t).
Consequently, the solution to the initial value
problem x' = Ax, x(0) = x0, is x = eAtx0.
THEOREM 6.5.3





Let A and B be n × n constant matrices and t, τ be
real or complex numbers. Then,
(a) eA(t+τ) = eAt eAτ.
(b) A commutes with eAt, that is, A eAt = eAt A.
(c) (eAt)−1 = e−At.
(d) e(A+B)t = eAt eBt if AB = BA, that is, if A and B
commute.
Methods for Constructing eAt

If a fundamental set of solutions
to x' = Ax exists then by Theorem 6.5.2
eAt = X(t)X−1(0),
where
Example
Answer
eAt When A Is Nondefective.
In the case that A has n linearly independent
eigenvectors {v1, . . . , vn}, then, by Theorem
6.3.1, a fundamental set is {eλ1tv1, eλ2tv2, . . .
, eλntvn}.
 Then


Where
Example
Using the Laplace Transform to
Find eAt .
Apply the method of Laplace transforms to
the matrix initial value problem Φ' = AΦ,
Φ(0) = In.
 We denote the Laplace transform of Φ(t) by


We can then recover Φ(t) = eAt by taking the
inverse Laplace transform of the expression
on the right-hand side of
eAt = L−1 {(sIn − A)−1}(t).
Example
Answer
6.6 Nonhomogeneous Linear
Systems
Variation of Parameters
In this section, we turn to the nonhomogeneous
system
x' = P(t)x + g(t),
(1)
where the n × n matrix P(t) and the n × 1 vector g(t)
are continuous forα <t <β. Assume that a
fundamental matrix X(t) for the corresponding
homogeneous system
x' = P(t)x
(2)
has been found. We use the method of variation of
parameters to construct a particular solution, and
hence the general solution, of the
nonhomogeneous system (1).

General Solution

Even if the integral cannot be evaluated, we
can still write the general solution of Eq. (1) in
the form

With the initial condition x(t0) = x0
the general solution of the differential equation
is
 Using the fundamental matrix Φ(t) satisfying
Φ(t0) = In we have

The Case of Constant P.
If the coefficient matrix P(t) in Eq. (1) is a
constant matrix, P(t) = A, it is natural and
convenient to use the fundamental matrix
Φ(t) = eAt to represent solutions to x' = Ax +
g(t).
 The general solution takes the form


If an initial condition is prescribed at t = t0 is
x(t0) = x0, then c = e−At0x0 and we get
Example

Answer
Undetermined Coefficients and
Frequency Response
The method of undetermined coefficients,
discussed in Section 4.5, can be used to
find a particular solution of
x' = Ax + g(t)
if A is an n × n constant matrix and the
entries of g(t) consist of polynomials,
exponential functions, sines and cosines, or
finite sums and products of these functions.
The methodology described in Section 4.5
extends in a natural way to these types of
problems and is discussed in the exercises
(see Problems 14–16).

Example

Consider the circuit shown in Figure that was
discussed in Section 6.1. Using the circuit
parameter values L1 = 3/2, L2 = 1/2, C = 4/3,
and R = 1, find the frequency response and
plot a graph of the gain function for the output
voltage vR = Ri2(t) across the resistor in the
circuit.
Answer

It follows from G(iω) = −(A − iωI3)−1B that the
frequency response of the output voltage vR
= Ri2(t) is given by RG2(iω) = ( 0 R 0 )G(iω).
6.7 Defective Matrices

Fundamental Sets for Defective Matrices.

Example
Answer

Following Theorem 6.7.1, we find the
fundamental set

and
Example
Answer: Using complex arithmetic and Euler’s formula,
we find the following fundamental set of four real-valued
solutions of Eq. (13):
CHAPTER SUMMARY

Section 6.1 Many science and
engineering problems are modeled by
systems of differential equations of
dimension n > 2: vibrating systems
with two or more degrees of freedom;
compartment models arising in
biology, ecology, pharmacokinetics,
transport theory, and chemical reactor
systems; linear control systems; and
electrical networks.
Section 6.2


If P(t) and g(t) are continuous on I, a unique
solution to the initial value problem x' = P(t)x + g(t),
x(t0) = x0, t0 ∈ I exists throughout I.
A set of n solutions x1, . . . , xn to the
homogeneous equation x' = P(t)x, P continuous on
I is a fundamental set on I if their Wronskian
W[x1, . . . , xn](t) is nonzero for some (and hence
all) t ∈ I . If x1, . . . , xn is a fundamental set of
solutions to the homogeneous equation, then a
general solution is x = c1x1(t) + ・・ ・+cnxn(t),
where c1, . . . , cn are arbitrary constants. The n
solutions are linearly independent on I if and only
if their Wronskian is nonzero on I.
Summary
Summary
Summary