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page 409 —————————————————————————— CHAPTER 7. —— I A It can be shown that the partial sums on the left hand side converge for all . Taking the
limit as
on both sides of the equation, we obtain A Alternatively, consider the system x
Ax . Since ODEs are uncoupled, the vectors
x
e
, are a set of linearly independent solutions. Hence
the matrix
X e e is a fundamental matrix. Finally, since X
e e I , it follows that e e 17 . Assuming that x
is a solution, then
both sides of the equation to obtain A.
A , with A x Integrate . Hence
x A . . Proceed with the iteration
x
With A . x , and noting that A is a constant matrix, ________________________________________________________________________
page 410 —————————————————————————— CHAPTER 7. —— x Ax x
That is, I Ax A x. . We then have
x AI x Ax
I A Ax
Ax A x. Now suppose that
I A A A A A x. It follows that
AI AI
A A A x A
A A A
A x
x Therefore
I A By induction, the asserted form of
. Define lim A A x. is valid for all . It can be shown that the...
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This note was uploaded on 03/11/2014 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 Staff

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