Analyzing Linear Systems with Complex Eigenvalues: A Case Study

[stunner5000pt]

For this linear system with complex eigenvalues
a) find the eigenvalues
b) determine whether the origin is a spiral source, sink or center
c) Determine the direction of oscillations, clockwise or anticlockwise

$$\frac{dY}{dt} = \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) Y$$ with initial conditions $$Y_{0} = (1,0)$$

i foudn the eigenvalues to be
$$\lambda = \pm i \sqrt{2}$$ which would make it a center
also the eigenvectors
$$\left(\begin{array}{cc}0&2\\-2&0\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = \pm i \sqrt{2} \left(\begin{array}{cc}x\\y\end{array}\right)$$ i computed to be
$$V_{1} = \left(\begin{array}{cc}i\sqrt{2}\\1\end{array}\right)$$
and $$V_{1} = -V_{2}$$

i feel i made a mistake in finding the eigenvectors
also what would be the direction of the oscillations then?? Do i solve the Initial value problem to get hte direction of the oscillations??

 

[HallsofIvy]

You have V₁ correct but V₂ is NOT -V₁.
$$V_{2} = \left(\begin{array}{cc}-i\sqrt{2}\\1\end{array}\right)$$

In order to determine the direction of rotation, look what happens to (1, 0):
dx/dt= 2y= 0 but dy/dt= -2 so the "motion" is downward and the rotation is clearly clockwise.

 

[stunner5000pt]

thank you very much didnt realize that the dx/dt and dy/dt were the directions of the vector. But what if the matrix was i nthe form
a b
c d then would i have to reduce this till i get zeroes in the 1x2 and 2x1 spots?