Science:Math Exam Resources/Courses/MATH307/April 2012/Question 07 (a)

MATH307 April 2012

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Question 07 (a)

Consider the MATLAB/Octave computation 1> A=[0 2 1; 1 0 0; 0 1 0] A = 0 2 1 1 0 0 0 1 0 2> [S D]=eig(A) S = 0.80902 -0.57735 0.30902 0.50000 0.57735 -0.50000 0.30902 -0.57735 0.80902 D = Diagonal Matrix 1.61803 0 0 0 -1.00000 0 0 0 -0.61803 (a) Write down a recursion relation for a sequence x0, x1, x2, x3, ... that you can analyse using this calculation. How many initial values do you have to specify to define the sequence?

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!

Hint

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can actually infer all the information we need to answer this question from the matrix A that was given. Let us first rewrite the matrix A in the context of how it is used: ${\begin{bmatrix}x_{n+1}\\x_{n}\\x_{n-1}\end{bmatrix}}={\begin{bmatrix}0&2&1\\1&0&0\\0&1&0\end{bmatrix}}{\begin{bmatrix}x_{n}\\x_{n-1}\\x_{n-2}\end{bmatrix}}$ This equation yields the following three linear equations: ${\begin{aligned}x_{n+1}&=2x_{n-1}+x_{n-2}\\x_{n}&=x_{n}\\x_{n-1}&=x_{n-1}\end{aligned}}$ The first equation of this set of linear equations tells us the recursive relationship for this matrix and notice that to calculate the value for the $n^{th}$ value in the relationship where $n>2$ and $n$ is an integer, we only need the first three values to be initially stated. In summary, the recursive sequence that is inferred from A is: $\displaystyle x_{n+1}=2x_{n-1}+x_{n-2}$ and we would need the first three values of the sequence to calculate all the following values in the sequence ($x_{0},x_{1},$ and $x_{3}$).

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