# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Example 2 is entirely typical of the general case when there is a double eigenvalue

and a single associated eigenvector. Consider again the system (1), and suppose that

r = p is a double eigenvalue of A, but that there is only one corresponding eigenvector g. Then one solution [similar to Eq. (9)] is

x(1)(t) = gept, (21)

where g satisfies

(A - pI)g = 0. (22)

By proceeding as in Example 2 we find that a second solution [similar to Eq. (19)] is

x(2)(t) = gtept + ^ept, (23)

where g satisfies Eq. (22) and ^ is determined from

(A - pI)-n = g. (24)

Even though det(A — pI) = 0, it can be shown that it is always possible to solve

Eq. (24) for ^. The vector ^ is called a generalized eigenvector corresponding to the

eigenvalue p.

Fundamental Matrices. As explained in Section 7.7, fundamental matrices are formed by arranging linearly independent solutions in columns. Thus, for example, a

406

Chapter 7. Systems of First Order Linear Equations

fundamental matrix for the system (8) can be formed from the solutions x(1)(t) and x(2)(t) from Eqs. (9) and (19), respectively:

w) = (_ee2t

te2t

-te2t-e2t

e

2t

1t

-1 -1-t

(25)

The matrix Ô that satisfies Ô(0) = I can also be readily found from the relation Ô({) = ^(t)^-1 (0). ForEq. (8) we have

^(0) =

1

0

11

^-1(0) =

1

0

11

(26)

and then

Ô(³) = ^(t)^-1(0) = e'

2t

e

2t

1 0 -1 -1-1/\-1 -1

1 t t

t 1 +1}'

The latter matrix is also the exponential matrix exp (At).

(27)

Jordan Forms. An n x n matrix A can be diagonalized as discussed in Section 7.7 only if it has a full complement of n linearly independent eigenvectors. If there is a shortage of eigenvectors (because of repeated eigenvalues), then A can always be transformed into a nearly diagonal matrix called its Jordan5 form, which has the eigenvalues of A on the main diagonal, ones in certain positions on the diagonal above the main diagonal, and zeros elsewhere.

Consider again the matrix A given by Eq. (2). Form the transformation matrix T with the single eigenvector g from Eq. (6) in its first column and the generalized eigenvector ^ from Eq. (17) with k = 0 in the second column. Then T and its inverse are given by

T

1

0

11

As you can verify, it follows that

T-1AT =

T- 1

2 1 0 2

1

0

11

J.

(28)

(29)

The matrix J in Eq. (29) is the Jordan form of A. It is typical of all Jordan forms in that it has a 1 above the main diagonal in the column corresponding to the eigenvector that is lacking (and is replaced in T by the generalized eigenvector).

If we start again from Eq. (1),

x' = Ax,

the transformation x = Ty, where T is given by Eq. (28), produces the system

y = Jy, (30)

where J is given by Eq. (29). In scalar form the system (30) is

/1 = 2 / + />> Ó2 = 2 /2 • (31)

Camille Jordan (1838-1921), professor at the Ecole Polytechnique and the College de France, made important contributions to analysis, topology, and especially to algebra. The Jordan form of a matrix appeared in his influential book Traite des substitutions et des equations algebriques, published in 1870.

7.8 Repeated Eigenvalues 407

These equations can be solved readily in reverse order. In this way we obtain

C2

/2 = C1e2t > /1 = C1te2t + C2e2t. (32)

Thus two independent solutions of the system (30) are

/<¦>(,) = (0) e2', y® = (JJ e2' (33)

and the corresponding fundamental matrix is

Ô(t) = (^ ^) . (34)

Since Ô (0) = I, we can also indentify the matrix in Eq. (34) as exp (Jt). The same result can be reached by calculating powers of J and substituting them into the exponential series (see Problems 19 through 21). To obtain a fundamental matrix for the original system we now form the product

9(t) = T exp(Jt) = (—^ — te2<) , (35)

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