Clockwise or counterclockwise? (linear system phase portrait)

[ak416]

Homework Statement

Given a 2x2 matrix A with entries a,b,c,d (real) with complex eigenvalues I would like to know how to find out whether the solutions to the linear system are clockwise or counterclockwise. (Some kind of inequality between a,b,c,d).

Homework Equations

The Attempt at a Solution

I tried looking at the signs of each component of the derivative. It seems to me that clockwise means x1' > 0 and x2' < 0 for x1,x2 > 0 as long as x1 or x2 are not too small. Then I am not sure...

 

[HallsofIvy]

The simplest way is to see what the matrix does to (1, 0) and (0, 1).

For example, if the problem is
$$\frac{dX}{dt}= \left( \begin{array}{cc}0 & -1\\ 1 & 0\end{array}\right)X$$
Then
$$\left( \begin{array}{cc}0 & -1\\ 1 & 0\end{array}\right)\left(\begin{array}{c} 1 \\ 0\end{array}\right)= \left(\begin{array}{c}0 \\ 1\end{array}\right)$$
That's counter-clockwise rotation. The matrix
$$\left(\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right)$$
has exactly the same eigenvalues but is clockwise rotation.