6.5 Fundamental Matrices and the Exponential of a Matrix

1. 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.

2. Example Question: Find a fundamental matrix for the system x' = x. Answer:

3. 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

4. Example For the system in previous Example, find the fundamental matrix Φ such that Φ(0) = I2. Answer:

5. 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:

6. 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= eAtA. (c) (eAt)−1 = e−At. (d) e(A+B)t = eAt eBtif AB = BA, that is, if A and B commute. THEOREM 6.5.3

7. 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

8. Example Answer

9. 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

10. Example

11. 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) = eAtby taking the inverse Laplace transform of the expression on the right-hand side of eAt= L−1 {(sIn− A)−1}(t).

12. Example Answer

13. 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).

14. 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

15. 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) = eAtto 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

16. Example Answer

17. 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).

18. 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.

19. 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).

20. 6.7 Defective Matrices Fundamental Sets for Defective Matrices. Example

21. Answer Following Theorem 6.7.1, we find the fundamental set and

22. Example Answer: Using complex arithmetic and Euler’s formula, we find the following fundamental set of four real-valued solutions of Eq. (13):

23. 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.

24. 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, . . . , xnto 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, . . . , xnis a fundamental set of solutions to the homogeneous equation, then a general solution is x = c1x1(t) + ・・ ・+cnxn(t), where c1, . . . , cnare arbitrary constants. The n solutions are linearly independent on I if and only if their Wronskian is nonzero on I.

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