Science:Math Exam Resources/Courses/MATH221/December 2007/Question Section 103 10 (a)
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Question Section 103 10 (a)
A discrete dynamical system is defined by
Find explicit formulas for and , with the given initial condition.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
Let us first write the system in matrix notation:
Notice that and
Therefore, we hypothesize that for every
This explicit formula can (and should) be established rigorously by induction (we leave this as an exercise).
Now we want to compute . Recall that matrix powers are easy to compute via matrix diagonalization. Therefore, we find the eigenvalues and eigenvectors for .
The eigenvalues are determined from , so the eigenvalues of A are 4 and -1.
To find an eigenvector associated with the eigenvalue 4, we solve , i.e. the equation
from which it is clear that is an eigenvector.
To find an eigenvector associated with the eigenvalue -1, we solve
from which we find is an eigenvector.
Now we form the matrix whose columns are the eigenvectors of A, and the diagonal matrix with the corresponding eigenvalues. We know that and hence
The final step is to compute , and then :
Check that for n = 1 we recover the matrix A.
Finally, the question asks us to find and explicitly, with the given initial condition. Having computed , we can now compute
Upon multiplying and simplifying the above equation, we find that