Science:Math Exam Resources/Courses/MATH307/December 2010/Question 05 (c)

MATH307 December 2010

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Question 05 (c)
Define a sequence x0, x1, ... by the initial conditions x0 = a, x1 = b and x2 = c together with the recursion relation $\displaystyle x_{n+3}=x_{n+2}+x_{n+1}+x_{n}$ for n = 0, 1, 2, ... (c) Explain how you could ensure that the a, b and c you find in part (b) are real numbers.

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
For a real matrix, if $\displaystyle v$ is an eigenvector to a complex eigenvalue $\displaystyle \lambda$, then $\displaystyle {\overline {v}}$ is an eigenvector to a complex eigenvalue $\displaystyle {\overline {\lambda }}$

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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. For real matrices, just as complex eigenvalues, also complex eigenvectors come in conjugate pairs. Hence $\displaystyle v_{2}={\overline {v_{3}}}$ Hence v2 + v3 is a real vector and therefore we ensure that the solution vector [c, b, a]T in part (b) is real by choosing real coefficients s and t and setting s = t: $\left\{{\begin{bmatrix}c\\b\\a\end{bmatrix}}{\text{ such that }}{\begin{bmatrix}c\\b\\a\end{bmatrix}}=s(v_{2}+v_{3}){\text{ for some }}s\in \mathbb {R} \right\}$

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