Finding eigenvectors for complex eigenvalues

[Dusty912]

Homework Statement

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

The Attempt at a Solution

using AY-λY=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 isn't (5, -2-i√11) one? they both work when I plug them in.

 

[Samy_A]

Homework Statement

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

The Attempt at a Solution

using AY-λY=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 isn't (5, -2-i√11) one? they both work when I plug them in.

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 $(5,-2-i\sqrt{11})$.

 

[Dusty912]

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

 

[Dusty912]

They appear to be scalar multiples of each other

 

[Samy_A]

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

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

They appear to be scalar multiples of each other

Indeed. So either answer is correct.

 

[Dusty912]

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

 

[Dusty912]

computing the general solution with e^λt*V where V is the eigen vector

 

[Samy_A]

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

computing the general solution with e^λt*V where V is the eigen vector

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