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

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 x_n converges to

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

as $n\to \infty$.

(b) What is the value of the inner (dot) product $\langle \mathbf {x} _{\infty },A\mathbf {x} _{\infty }\rangle$?