- #1

junsugal

- 6

- 0

## Homework Statement

This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example.

We will define two sequences of vectors{v

_{k}} and {u

_{k}}

(a) Choose any vector u [itex]\in[/itex] R

^{2}as u

_{0}

(b) Once u

_{k}has been determined, the vectors v

_{k+1}and u

_{k+1}are determined as follows:

i. Set v

_{k+1}= Au

_{k}

ii. Find the entry of maximum absolute value of v

_{k+1}let's say it is the j-th entry of v

_{k+1}

iii. Set u

_{k+1}= v

_{k+1}/v

_{jk+1}

(c) The sequence {U

_{k}} will converge to an eigenvector for A.

Let A =

2 1

1 2

Use the above method to approximate an eigenvector for A using only 4 iterations, that is finding v

_{5}.

Find the eigenvectors for A using the method learned in class and compare.

## Homework Equations

## The Attempt at a Solution

I assume u

_{0}is (1,1)

From A, i found the eigenvalues which is 3 and 1.

And the corresponding eigenvectors is (1,1) and (-1,1)

Then i get u

_{k}as (3

^{k}-1, 3

^{k}+1)

Now, to the next step, I get my v

_{k+1}as ( 2*(3

^{k}-1)+3

^{k}+1, 3

^{k}-1 + 2*(3

^{k}+1))

Plugging in k as 4 as I want to get v

_{5}

I get v

_{5}as (242, 244)

And from this step the instruction said to find the entry of maximum absolute value of v

_{k+1}and I don't know how. I am not even sure if what I've done is correct.

Please correct me if I am wrong and please explain on what to do next.

Thanks in advance!