Finding eigenvalues and eigenvectors of a matrix A

  • Thread starter Rubik
  • Start date
  • #1
97
0

Homework Statement



Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4]

Homework Equations



det(A - lambda I) = 0

The Attempt at a Solution



After expanding using the formula I have the equation (2 - [itex]\lambda[/itex])(4 - [itex]\lambda[/itex]) - 8

Which gives [itex]\lambda[/itex] = 0, 6 (Should I be worried that I got a zero in my answer?)

Then substituting the values for lambda back into the problem and using (A - [itex]\lambda[/itex]I)v = 0

For [itex]\lambda[/itex] = 6
[-4, 1; 8, -2] * [a; b] = [0; 0] and I get the eigenvector a[1; 4] from (-4a + b = 0)

For [itex]\lambda[/itex] = 0
[2, 1; 8, 4] * [a; b] = [0; 0] I get a second eigenvector of a[1; -2] from (2a + b = 0)

Did I get the right eigenvalues and corresponding eigenvectors?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
619
Yes, you did. And it's easy for you to check as well. A*[1;4] should give you 6*[1;4] and A*[1;-2] should give you 0*[1;-2], right? Does it work?
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
964

Homework Statement



Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4]

Homework Equations



det(A - lambda I) = 0

The Attempt at a Solution



After expanding using the formula I have the equation (2 - [itex]\lambda[/itex])(4 - [itex]\lambda[/itex]) - 8

Which gives [itex]\lambda[/itex] = 0, 6 (Should I be worried that I got a zero in my answer?)
Notice that the determinant of the matrix is 2(4)- 1(8)= 0. Since the determinant of a matrix is the product of its eigenvalues, 0 must be an eigenvalue.

Then substituting the values for lambda back into the problem and using (A - [itex]\lambda[/itex]I)v = 0

For [itex]\lambda[/itex] = 6
[-4, 1; 8, -2] * [a; b] = [0; 0] and I get the eigenvector a[1; 4] from (-4a + b = 0)

For [itex]\lambda[/itex] = 0
[2, 1; 8, 4] * [a; b] = [0; 0] I get a second eigenvector of a[1; -2] from (2a + b = 0)

Did I get the right eigenvalues and corresponding eigenvectors?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Related Threads on Finding eigenvalues and eigenvectors of a matrix A

Replies
26
Views
3K
Replies
5
Views
765
Replies
15
Views
6K
Replies
2
Views
29K
Replies
6
Views
2K
Replies
5
Views
10K
Replies
10
Views
2K
Replies
1
Views
5K
Replies
4
Views
1K
Replies
5
Views
1K
Top