# Finding eigenvectors for complex eigenvalues

1. Apr 18, 2016

### Dusty912

1. The problem statement, all variables and given/known data
So I have been having trouble with finding the proper eigen vector for a complex eigen value
for the matrix A=(-3 -5)
. .........................(3 1)

had a little trouble with formating this matrix sorry
The eigen values are -1+i√11 and -1-i√11
3. The attempt at a solution
using AYY=0 (where Y is a vector (x, y) I obtain two different equations :
3x+y-(-1+i√11)y=0 and -3x-5y-(-1+i√11)x=0

these simplify too: 3x=(i√11 -2)y and 5y=(-2-i√11)x

no I know that the right eigen vecotor is (i√11 -2,3) but why isnt (5, -2-i√11) one? they both work when I plug them in.

2. Apr 18, 2016

### Samy_A

If $\vec x$ is an eigenvector of $A$ with eigenvalue $\lambda$, then so is $k \vec x$, where $k \in \mathbb C, k\neq 0$.
Check if $(i\sqrt {11}-2,3)$ is a (complex) scalar multiple of $(5,-2-i\sqrt{11})$.

3. Apr 18, 2016

### Dusty912

well are you saying that both of these are eigen vectors for this matrix?

4. Apr 18, 2016

### Dusty912

They appear to be scalar multiples of each other

5. Apr 18, 2016

### Samy_A

If they are non zero scalar multiples of each other, yes. As said, any non zero scalar multiple of an eigenvector is an eigenvector.
Indeed. So either answer is correct.

6. Apr 18, 2016

### Dusty912

but when i try to compute the general solution it looks different.

7. Apr 18, 2016

### Dusty912

computing the general solution with eλt*V where V is the eigen vector

8. Apr 18, 2016

### Samy_A

It may be me, but I don't understand what you are saying here.