Unformatted text preview: ition of an eigenvector, x x
, which implies that is real.
33. From Prob. x Hence we must have ,
Ax x x Ax . Hence
x x x x x x , since the eigenvalues are real. Therefore
Given that , we must have x x x .
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3. Eq. states that the Wronskian satisfies the first order linear ODE The general solution is
in which is an arbitrary constant. Let X and X be matrices representing two sets
of fundamental solutions. It follows that
Hence X X . Note that 4. First note that As shown in Prob.
x , x For second order linear ODE, the Wronskian as defined in Chap.
order differential equation
. It follows that satisfies the first Alternatively, based on the hypothesis, Direct calculation shows that Here we used the fact that . Hence
x 5. The particular solution satisfies the ODE x P x x
g Now let ________________________________________________________________________
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- Spring '08
- Linear Algebra, eigenvector