Tutorial 3 - practice exercises week 3 - Eigenvalues and eigenvectors PDF

Preview

Summary

practice exercises week 3 - Eigenvalues and eigenvectors...

Description

MATH1180 – Computational Methods and Numerical Techniques Tutorial Teaching Week 03 – Eigenvalues and eigenvectors Part 1 (Essential exercises) Q1) Calculate the eigenvalues and corresponding eigenvectors for each of the following matrices. (i)

 2 1    1 2

(ii)

 4 2    3 3

(iii)

5 7     3 25 

(iv)

0 2    1 0 

(v)

2 1     4 2

(vi)

 2 2    − 1 − 1

Q2) Show that the following matrices do not have two distinct real eigenvalues. (i)

 1 1    0 1

(ii)

 1 4   −3 2 

(iii)

0 0    0 0 

6  6  has distinct eigenvalues 2 and 3. Demonstrate by direct Q3) It is known that A =   − 2 − 1 calculation that A2 has eigenvalues 22 = 4 and 32 = 9. Without direct calculation find the eigenvalues of ฀฀5 . Q4) Calculate the eigenvalues and eigenvectors of the following 3 x 3 matrices – you will need to be able to solve cubic equations!

(i)

1 −1 4     3 2 − 1  2 1 − 1  

(ii)

 1 −1 0     − 1 2 − 1  0 −1 1   

Q5) (i) What can you conclude about A if λ = 0 is an eigenvalue for A? (ii) What are the eigenvalues and eigenvectors of an identity matrix? (iii) In each of the matrices of Q1 find the eigenvalues of its inverse and transpose using eigenvalue properties. Verify that the sum and product of the eigenvalues of each of the matrices satisfy the trace and the determinant of the matrix. (iv) Modify the spreadsheet from the Moodle site and obtain the largest eigenvalue of each of the matrices in Q1 using Power Method.

Part 2 (Individual exercises) Take the last three non-zero digits of your enrolment number a, b and c, and use these to create  a b a 2 x 2 matrix   . For example, with enrolment number 000654032, a = 4, b = 3 and c = 2, b c  4 3 and so you would create the matrix   .  3 2 1. Find the eigenvalues and eigenvectors of your matrix. 2. Verify all the eigenvalue properties as described in the lecture notes. 3. Use the spreadsheet for Power Method from the Moodle site to compute the largest eigenvalue and its corresponding eigenvector....