# complex eigenvalues

Discussion in 'Math' started by kokkie_d, Dec 3, 2010.

1. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
72
0
Hi

I have a matrix A which results in the eigenvalues:
$\lambda = -0.5 + j$ and its complex conjugate $\bar{\lambda} = -0.5 - j$
The eigen vectors are:
$v = 1 + j$ and its complex conjugate $\bar{v} = 1-j$
In state space the following formula is used:
$
x(t) = x_1 + x_2
x(t) = C_1 \left[\stackrel{1}{j}\right] e^{(-0.5+j) t} + C_1 \left[\stackrel{1}{-j}\right] e^{(-0.5-j) t}
$

Euler states:
$e^{(-0.5+j)t} = e^{-0.5t}(cos (t) + j sin(t))$
Ignoring $C_1 and C_2$ for now and multiplying with the eigenvectors:
$x_1a = e^{-0.5t}(cos (t) + j sin (t))$
$x_1b = e^{-0.5t}(-sin (t) + cos (t))$
$x_2a = e^{-0.5t}(cos (t) - j sin (t))$
$x_2b = e^{-0.5t}(sin (t) - j cos (t))$

I think I am correct thus far, please correct me if I am wrong.

The next steps should result in the following equation:

$x(t) = e^{-0.5t}(C_1\left[\stackrel{cos (t)}{-sin (t)} \right] + C_2\left[\stackrel{sin (t)}{cos (t)} \right])
$

But I can not seem to get there. I assume it has something to do with the complex eigenvectors and that I am not using them right. I hope someone can explain it to me.