Science:Math Exam Resources/Courses/MATH307/December 2010/Question 07 (c)

MATH307 December 2010

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Question 07 (c)

Suppose that A is a symmetric 4x4 matrix with eigenvalues 0, 1, 4, 5. Define a sequence of vectors $\mathbf {x} _{n}\in \mathbb {R} ^{4}$ by choosing x0 at random, and then settings ${\begin{aligned}\mathbf {y} _{n}&=(A-3I)^{-1}\mathbf {x} _{n-1}\\\mathbf {x} _{n}&=\mathbf {y} _{n}/\|\mathbf {y} _{n}\|\end{aligned}}$ for n = 1, 2, .... You then observe that xn converges to $\mathbf {x} _{\infty }=[1/2,1/2,1/2,1/2]^{T}$ as $n\to \infty$. (c) What vector does $\mathbf {y} _{n}$ converge to?

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
Take the limit $n\to \infty$ on both sides of the definition of $\mathbf {y} _{n}$.

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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. Taking the limit n → ∞ on both sides yields ${\begin{aligned}\lim _{n\to \infty }\mathbf {y} _{n}&=\lim _{n\to \infty }(A-3I)^{-1}\mathbf {x} _{n-1}\\&=(A-3I)^{-1}\lim _{n\to \infty }\mathbf {x} _{n-1}\\&=(A-3I)^{-1}\mathbf {x} _{\infty }\\&=\eta _{3}\mathbf {x} _{\infty }\\&=\mathbf {x} _{\infty }=[1/2,1/2,1/2,1/2]^{T}\end{aligned}}$ We used, from part (a), that $\mathbf {x} _{\infty }$ is an eigenvector of $\displaystyle (A-3I)^{-1}$ with eigenvalue $\eta _{3}=1$.

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