6.5 Fundamental Matrices and the Exponential of a Matrix Fundamental Matrices Suppose that x 1 (t),..., x n (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 x 1 (t),..., x n (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 x' = x. 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 x 1 (t),..., x n (t) designated in Theorem For initial condition, x(t 0 ) = x 0, 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) = I 2. Answer:

The Matrix Exponential Function e At DEFINITION Let A be an n × n constant matrix. The matrix exponential function, denoted by e At, is defined to be Example:

THEOREM If A is an n × n constant matrix, then e At = Φ(t). Consequently, the solution to the initial value problem x' = Ax, x(0) = x 0, is x = e At x 0. Let A and B be n × n constant matrices and t, τ be real or complex numbers. Then, (a) e A(t+τ) = e At e Aτ. (b) A commutes with e At, that is, A e At = e At A. (c) (e At ) −1 = e −At. (d) e (A+B)t = e At e Bt if AB = BA, that is, if A and B commute. THEOREM 6.5.3

Methods for Constructing e At If a fundamental set of solutions to x' = Ax exists then by Theorem e At = X(t)X −1 (0), where

Example Answer

e At When A Is Nondefective. In the case that A has n linearly independent eigenvectors {v 1,..., v n }, then, by Theorem 6.3.1, a fundamental set is {e λ1t v 1, e λ2t v 2,..., e λnt v n }. Then Where

Example

Using the Laplace Transform to Find e At. Apply the method of Laplace transforms to the matrix initial value problem Φ' = AΦ, Φ(0) = I n. We denote the Laplace transform of Φ(t) by We can then recover Φ(t) = e At by taking the inverse Laplace transform of the expression on the right-hand side of e At = L −1 {(sI n − 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(t 0 ) = x 0 the general solution of the differential equation is Using the fundamental matrix Φ(t) satisfying Φ(t 0 ) = I n 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) = e At to represent solutions to x' = Ax + g(t). The general solution takes the form If an initial condition is prescribed at t = t 0 is x(t 0 ) = x 0, then c = e −At0 x 0 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 L 1 = 3/2, L 2 = 1/2, C = 4/3, and R = 1, find the frequency response and plot a graph of the gain function for the output voltage v R = Ri 2 (t) across the resistor in the circuit.

Answer It follows from G(iω) = −(A − iωI 3 ) −1 B that the frequency response of the output voltage vR = Ri 2 (t) is given by RG 2 (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(t 0 ) = x 0, t 0 ∈ I exists throughout I. A set of n solutions x 1,..., x n to the homogeneous equation x' = P(t)x, P continuous on I is a fundamental set on I if their Wronskian W[x 1,..., x n ](t) is nonzero for some (and hence all) t ∈ I. If x 1,..., x n is a fundamental set of solutions to the homogeneous equation, then a general solution is x = c 1 x 1 (t) + ・・ ・ +c n x n (t), where c 1,..., c n are arbitrary constants. The n solutions are linearly independent on I if and only if their Wronskian is nonzero on I.

Summary