6.5 Fundamental Matrices and theExponential 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.

ExampleQuestion: Find a fundamental matrix for the

system

x' = x.

Answer:

14

11

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

ExampleFor 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.

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.

THEOREM 6.5.3

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 ParametersIn 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 SolutionEven 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).

ExampleConsider 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.

AnswerIt 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 MatricesFundamental Sets for Defective Matrices.

Example

AnswerFollowing 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