1. Apr 29, 2005

### djinteractive

I am having trouble getting started with this problem.. I guess I'll explain it first.

Find one possible matrix A for which the solution to Y'=AY with initial condition Y(0)=(6,13,9) has the following property. As time progresses, the solution Y(t) spirals toward the plane 2x+3y+4z=0 where it continues to circultate about a radius five circle.

I know I have to find some vectors Vr(real) and Vi(imaginary) that span the plane and are orthogonal to it. The eigen vectors if I remember correctly should come in a pair Vr + iVi and Vr-iVi that I can normalize with PDP-1(inverse) but I am having a problem with how to get these eigenvalues that span the plane.. will someone PLEASE (I'm begging you) help me.. this is driving me insane
PS the 3rd (in z direction should just be a real lambda value I think)

2. Apr 30, 2005

### kleinwolf

A way to attack the problem could be :

Take a vector in the plane : a=(2,0,-1)/Sqrt(5), the normal vector to the plane : n=(2,3,4) and a vector in the plane, orthogonal to the first : b=(3,-10,6)/Sqrt(145).

Then you notice that you're LUCKY : you can easily compute the distance of Y(0) to the direction given by n : it is 5.

Hence the curve is not a 3d spiral but just a screw type one !!

Hence you know that in the (a,b,n) basis, your vector expresses as :

x(t)=(5*cos(wt),5*sin(wt),A*exp(-Dt))

hence in the (a,b,n) basis : $$x'(t)=\left(\begin{array}{ccc} 0& -w & 0\\ w & 0 & 0\\0&0&-D\end{array}\right)x(t)$$

then you just have to change the basis to the canonical one and fit the initial condition.