Page 158 - Schaum's Outline of Differential Equations

CHAP. 16] e At 141

Note that Theorem 16.2 involves the eigenvalues of A?; these are t times the eigenvalues of A. When com-
puting the various derivatives in (16.7), one first calculates the appropriate derivatives of the expression (16.5)
with respect to A, and then substitutes A = \. The reverse procedure of first substituting A = A, (a function of f)
into (16.5), and then calculating the derivatives with respect to t, can give erroneous results.

Example 16.2. Let A have four rows and four columns and let A = 5t and A = 2t be eigenvalues of At of multiplicities
three and one, respectively. Then n = 4 and

5t
5t
5t
Since A = 5t is an eigenvalue of multiplicity three, it follows that e = r(5t), e = r'(5t), and e = r"(5t). Thus,

2t
Also, since A = 2t is an eigenvalue of multiplicity one, it follows that e = r(2t), or

Notice that we now have four equations in the four unknown a's.
Method of computation: For each eigenvalue A,, of A?, apply Theorem 16.2 to obtain a set of linear
equations. When this is done for each eigenvalue, the set of all equations so obtained can be solved for a 0,
At
«i, ... , «„_!. These values are then substituted into Eq. (16.2), which, in turn, is used to compute e .

Solved Problems

At
16.1. Find e for A =
Here n = 2. From Eq. (16.3),

and from Eq. (16.5), r(A) = a{k + OQ. The eigenvalues of At are A^ = 4t and A^ = —2t, which are both of multiplicity
one. Substituting these values successively into Eq. (16.6), we obtain the two equations

Solving these equations for c^ and OQ, we find that

and

Substituting these values into (1) and simplifying, we have