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Finding the eigenvectors?
For the matrix M=
-e 0 d
e -n 0
0 n -d
I wrote the characteristic equation and found that one of the eigenvalues is zero and I'm trying to find the corresponding eigenvector s. So I wrote Ms=0 but then I'm not sure what the eigenvector is because each equation gives information including only two variables: so
from the 1st row we get d*z=e*x (with no information about y)
from the 2nd row we get e*x=n*y (with no information about z)
from the 3rd row we get n*y=d*z (with no information about x)
so.... What is the eigenvector?
Any help would be greatly appreciated.
Thanks very much!
Hi thanks for your reply Johnny. I don't really understand why you had to get M into rref and how you deduced that the eigenvector was the zero vector. If we follow the method where once finding that the eigenvalue x=0 we write:
Ms=0*s=0
-e 0 d ] [X]
e -n 0 ] [Y] = *0*
0 n -d ] [Z]
so then from the first row you get the vector
d
?
e
is the place of the question mark supposed to be zero?
1 Answer
- 7 years ago
Recall, to find the eigenvalues we solve det( M - xI) = 0, where M is the matrix given, x is the eigenvalue, and I is the identity matrix:
M - xI yields:
- e - x..........0............d
.....e.......- n - x..........0
.....0............n.......- d - x
Find the determinant of this, I'll be using cofactor expansion using the 1st row, which gives:
( - e - x ) | - n - x.........0 ...| + d | e.....- n - x |
...............| ....n.......- d - x .|.......| 0..........n ..|
( - e - x ) ( - n - x ) ( - d - x ) + d ( en )
( - d - x ) ( en + xe + xn + x^2 ) + den
- den - xde - xdn - dx^2 - xen - ex^2 - nx^2 - x^3 + den
- ( x^3 + ( d + e + n ) x^2 + ( de + dn + en ) x )
- x ( x^2 + ( d + e + n ) x + de + dn + en )
Setting this equal to 0 and solving, you're right, you do get an eigenvalue of 0, but that's only 1 eigenvalue, be sure you can solve for the others.
To get the eigenvector, substitute x = 0 into M - xI = 0 and solve it.
M - ( 0 ) I = 0
M = 0
So, we basically just have to get M into rref and get our solution set. This is the eigenvector for 0.
Add row 1 and row 2 together, replace row 2:
- e.....0.....d
.0...- n.....0
.0.....n...- d
Add row 2 and row 3 together, replace row 3:
- e.....0.....d
.0...- n.....0
.0.....0...- d
Add row 3 and row 1 together, replace row 1:
- e.....0.....0
.0...- n.....0
.0.....0...- d
Divide each row by the appropriate value, - e, - n, and - d...and you'll end up with the identity matrix.
So you're eigenvector is actually just the 0 vector.
Sort of a weird problem...
Hope that helps!
