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whose columns are g(1),..., g(n). Then, defining a new dependent variable y by the
x = Ty, (38)
we have from Eq. (37) that
Ty' = ATy.
7.7 Fundamental Matrices
Multiplying by T ',we then obtain
y = (T—!at )y,
or, using Eq. (32),
y' = Dy.
Recall that D is the diagonal matrix with the eigenvalues rv ..., rn of A along the diagonal. A fundamental matrix for the system (41) is the diagonal matrix (see Problem 16)
Q(t) = exp(Dt) =
ZeV 0 . .0
0 & . 0
0 0 ¦ ¦ ernl)
A fundamental matrix ^ for the system (37) is then found from Q by the transformation (38),
Equation (44) is the same result that was obtained in Section 7.5. This diagonalization procedure does not afford any computational advantage over the method of Section 7.5, since in either case it is necessary to calculate the eigenvalues and eigenvectors of the coefficient matrix in the system of differential equations. Nevertheless, it is noteworthy that the problem of solving a system of differential equations and the problem of diagonalizing a matrix are mathematically the same.
Consider again the system of differential equations
x' = Ax,
where A is given by Eq. (33). Using the transformation x = Ty, where T is given by Eq. (35), you can reduce the system (45) to the diagonal system
y'= 0 -1) y = Dy.
Obtain a fundamental matrix for the system (46) and then transform it to obtain a fundamental matrix for the original system (45).
By multiplying D repeatedly with itself, we find that
Chapter 7. Systems of First Order Linear Equations
Therefore it follows from Eq. (23) that exp(Dt) is a diagonal matrix with the entries e3t and e-t on the diagonal, that is,
<•”' = (e0 e-0) • (48)
Finally, we obtain the required fundamental matrix ^(t) by multiplying T and exp(Dt):
*(t)=(2 JHeo e-0)=-³ª-t) (49»
Observe that this fundamental matrix is the same as the one found in Example 1.
PROBLEMS In each of Problems 1 through 10 find a fundamental matrix for the given system of equations. = In each case also find the fundamental matrix Ô(´) satisfying Ô(0) = I.
1. x' =
7. x' =
1 _ 3
:3 -2)x 4 x=(4 -1
^2 -2)x 6. x'=(-1 -4
'5 -0x 8. x'=G -3
-2)x 2. x' =
-1) x 4. x' =
-5)x 6. x' =
-0x 8. x' =
1 -1 1 x 10. x' =
4 -1 4\
9. x' = I 2 1 -1 I x 10. x' = I 3 2 — 1 I x
11. Solve the initial value problem
3 -2)x x(0)=(-2
by using the fundamental matrix Ô(¥) found in Problem 3.
12. Solve the initial value problem
x'=(-1 -0x, x(0)=(3
by using the fundamental matrix Ô(¥) found in Problem 6.
13. Show that Ô^) = 'Ô'Þ'Ô'-1 (t0), where Ô1t) and ^(t) are as defined in this section.
14. The fundamental matrix Ô(¥) for the system (3) was found in Example 2. Show that Ô(0Ô^) = Ô(´ + s) by multiplying Ô(0 and Ôis).
15. Let Ô(´) denote the fundamental matrix satisfying Ô' = AÔ, Ô(0) = I. In the text we also denoted this matrix by exp (A t). In this problem we show that Ô does indeed have the principal algebraic properties associated with the exponential function.
(a) Show that Ôit)Ôis) = Ô^ + s); that is, exp(At) exp(As) = exp[A(t + s)].
Hint: Show that if s is fixed and t is variable, then both Ôit)Ôis) and Ô^ + s) satisfy the initial value problem Z' = AZ, Z(0) = Ôis).
(b) Showthat Ô^)Ô(-´) = I;thatis, exp(At) exp[A(-1)] = I. Then show that Ô(-1) =
(c) Show that Ô (t - s) = Ô (´)Ô 1 (s).
7.8 Repeated Eigenvalues
16. Show that if A is a diagonal matrix with diagonal elements a1, a2,..., an, then exp (At) is also a diagonal matrix with diagonal elements exp(a1t), exp(a21), ..., exp (at).
17. The method of successive approximations (see Section 2.8) can also be applied to systems of equations. For example, consider the initial value problem
x' = Ax, x(0) = x0, (i)