MATH307 April 2006
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Question 02 (a)
Consider the diagonalizable matrix
(a) Find a solution of the difference equation satisfying the initial condition . Determine the limit of the ratio as .
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We want to find the ratio of as for the difference equation . The difference equation suggests that for every increment of 1 unit time, we want to multiply by .
The matrix A is already given in the diagonalized form of
Given the initial condition of , we have the equation , which can also be expressed as the form
where you can obtain c1, c2 by plugging in the initial condition x0
As , the component with would reach 0 and you would be left with
and therefore the ratio of would just be .