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

MATH307 April 2012

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[hide] Question 07 (c)
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 (c) Write down initial values which result in a non-zero periodic sequence where the xn repeatedly cycle through the same values.

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[show] 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. Let us begin by assuming that the cycle only repeats over a single value. It should be immediately obvious that the only possible value that this could be is 0 since if the initial values were less than 0, the values will become increasingly negative and if the initial values were greater than 0, the values will become increasingly positive. Since we want a non-zero periodic sequence, a cycle of 1 value is not possible. Therefore, we will look at a cycle which repeats over 2 values. This implies: ${\begin{aligned}x_{2}&=x_{0}\\x_{3}&=x_{1}\\&\vdots \\x_{n}&=x_{n-2}\end{aligned}}$ If we just take the first value that can use the recursion relation ($x_{3}$), we see that we can arrive at the following through basic algebra (where in step two we use that $x_{3}=x_{1}$ : ${\begin{aligned}&x_{3}=2x_{1}+x_{0}\\&x_{1}=2x_{1}+x_{0}\\&x_{1}+x_{0}=0\\&x_{1}=-x_{0}=-x_{2}\end{aligned}}$ We just showed that a recursive cycle of 2 values is possible! If we assign the first two values to be the negative of each other, we will have a repeating cycle of the two selected values. Therefore, the following are all possible choices for the initial three values (remember that we need three initial values!): ${\begin{bmatrix}x_{0}\\x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}1\\-1\\1\end{bmatrix}},{\begin{bmatrix}-1\\1\\-1\end{bmatrix}},{\begin{bmatrix}2\\-2\\2\\\end{bmatrix}},{\begin{bmatrix}-\pi \\\pi \\-\pi \\\end{bmatrix}},etc.$