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

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MATH307 December 2010
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Question 07 (a)

Suppose that ${\displaystyle A}$ is a symmetric ${\displaystyle 4\times 4}$ matrix with eigenvalues 0, 1, 4, 5. Define a sequence of vectors ${\displaystyle \mathbf {x} _{n}\in \mathbb {R} ^{4}}$ by choosing ${\displaystyle \mathbf {x} _{0}}$ at random, and then settings

{\displaystyle {\begin{aligned}\mathbf {y} _{n}&=(A-3I)^{-1}\mathbf {x} _{n-1}\\\mathbf {x} _{n}&=\mathbf {y} _{n}/\|\mathbf {y} _{n}\|\end{aligned}}}

for ${\displaystyle n=1,2,\dots }$. You then observe that ${\displaystyle \mathbf {x} }$n converges to

${\displaystyle \mathbf {x} _{\infty }=[1/2,1/2,1/2,1/2]^{T}}$

as ${\displaystyle n\to \infty }$.

(a) What is ${\displaystyle A\mathbf {x} _{\infty }}$?

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