complex eigenvalues

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

  1. kokkie_d

    Thread Starter Active Member

    Jan 12, 2009

    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:
     <br />
x(t) = x_1 + x_2<br />
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}<br />
    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])<br />
    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.