Linear Algebra and Its Applications · Problem 7.3.1

Computations with Matrices — Gilbert Strang

Mathematics · power-method

Problem

For the matrix

A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}

with eigenvalues

\lambda_1 = 1

and

\lambda_2 = 3

, apply the power method

u_{k+1} = Au_k

three times to the initial guess

u_0 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}

. What is the limiting vector

u_\infty