NOTE: This is a homework problem (not graded). I want to learn how to do this math problem, not just be given an answer. Thank you for any help you can give!
So, in this problem I am dealing with trajectories of a system when eigenvalues are purely imaginary. It is stated in the homework problem that the trajectories are as follows:
(x)' (a11 a12)(x)
(y) = (a21 a22)(y)
*These are matrices.
Question A.) states:
Show that the eigenvalues are of the coefficient matrix are purely imaginary if and only if:
a11 + a22 = 0,
a11*a22 - a12*a21 > 0
My work for A.):
λ^2 - trace(A)λ + determinant(A);
λ^2 - (a11+a12)λ + (a11*a22 - a12*a21);
Next I find the zeros to solve for the eigenvalues. This proves that the eigenvalues MUST be imaginary (I believe). I use the quadratic formula, but ignore the division by 2*a because it is irrelevant in determining if the system is imaginary or not..
(a11+a22) ± √[(a11+a22)^2 - 4(1)(a11*a22 - a12*a21)]
We assume that a11 + a22 = 0 and a11*a22 - a12*a21 > 0
I now plug in the 0 for all a11 + a22:
0 ± √[(0)^2 - 4(1)(a11*a22 - a12*a21)]
± √[-4(a11*a22 - a12*a21)]
Now if a11*a22 - a12*a21 > 0, than it necessarily follows that the system's trajectory is imaginary.
Question B.) states:
dy/dx = (dy/dt)/(dx/dt) = (a21*x + a22*y)/(a11*x + a12*y)
Use a11 + a12 = 0 to show that this is exact.
For this question, do I purely plug in dx/dt and dy/dt, respectively? I am not sure how to prove this is 'exact'.
Question C.) states:
By integrating the last equations show that a12*x^2 + 2*a22*x*y + a12*y^2 = k where k is constant. Use the equations:
a11 + a22 = 0,
a11*a22 - a12*a21 > 0
to show that the graph is always an ellipse.
How can I show that this is ALWAYS an ellipse? Do I just integrate the system and work it until I come to a12*x^2 + 2*a22*x*y + a12*y^2 = k?