# Thread: eigenvector, complex roots

1. ## eigenvector, complex roots

I am asked to solve the system:

dx/dt =
-2 -2
4 2

with the initial condition
x(0) =
-1
-1

I find the roots of the characteristic equation to be +/- 2i
and the associated eigenvector to be
1
2-2i

from euler's formula, i find that x(t) = [cos(2t)+isin(2t)](1 : 2-2i)
x(t) =
cos2t + isin2t
2cos2t-2icos2t+2isin2t+2sin2t

x(t) =
cos2t
2cos2t + 2sin2t

+
(i)
sin2t
-2cos2t+2sin2t

with the initial condition, i find c1 = -3 and c2 = -5/2.

Where am I wrong (submitted answer is not correct)
Thanks, let me know if anything needs more explanatioin.

2. Originally Posted by plopony
I am asked to solve the system:

dx/dt =
-2 -2
4 2

with the initial condition
x(0) =
-1
-1

I find the roots of the characteristic equation to be +/- 2i
and the associated eigenvector to be
1
2-2i

from euler's formula, i find that x(t) = [cos(2t)+isin(2t)](1 : 2-2i)
x(t) =
cos2t + isin2t
2cos2t-2icos2t+2isin2t+2sin2t

x(t) =
cos2t
2cos2t + 2sin2t

+
(i)
sin2t
-2cos2t+2sin2t

with the initial condition, i find c1 = -3 and c2 = -5/2.

Where am I wrong (submitted answer is not correct)
Thanks, let me know if anything needs more explanatioin.
Firstly, the associated eigenvector is not
1
2i
It is
1
-1-i.

Now, letting your eigenvalue be $\omega=\alpha +\beta i$, and your eigenvector to be $w=z_1+z_2i$ , you have that
$
X(t)= c_1e^{\alpha t}(cos(\beta t)z_1-sin(\beta t)z_2) +c_2e^{\alpha t}(cos(\beta t)z_2+sin(\alpha t) z_1)$

Here Your $\alpha$ is zero, so he exponentials become one. Your $z_1 = 1,-1;z_2=0,-1$
Hence, substituting your values in you get
$x(t) = c_1 cos(2t)+c_2 sin(2t); y(t) =-c_1 cos(2t) +c_1sin(2t) -c_2cos(2t)-c_2sin(2t)$
Using x(0) =-1, y(0) =-1
$c_1=-1, c_2 =2$

Can you work through all the steps? I would write it in matrix form, but am unable to using latex, I don't know how. And am too busy to hunt down how. So I called the top line of the Matrix X(t) x(t) and the bottom line y(t). I have also skipped a fair amount of the calculations, but hopefully you will be able to work your way through yourself.

3. thanks for the reply, I appreciate it. I think since you point out i got the eigenvector wrong, i should be all set.