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Spiraling Confidence

  1. Apr 6, 2005 #1
    The past month or so we have been studing 2 dimensional spaces for their eigenvalues, vectors etc. and have shyed away from higher dimensional systems. These systems have included repeated, complex, and repeated eigenvalues, their phase portraits and the like. But we received this problem which is a little different than all of that. Well here's the problem.

    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 circulate about a circle with radius 5.

    We have been working nearly everything from a matrix or differential equation in to general solutions and phase planes but not really the other way around, and without the complication of a 3 dimensional system. I know a few things (I think) about this system 2 of the directions (x,y for example) contain a conjugate pair +/- bi ( i = imaginary number) of eigenvectors along with a real eigenvector that is negative and forces initial conditions to spiral in to the center rather than just circle in the original plane. Also the radius of 5 is dependant upon the initial condition. I also have a sneaking suspicion I may have to tilt this system directly on it's axis and then renormalize it using PDP-1 (P-1 is supposed to be P inverse but don't know the tag for superscript) but I am unsure of this. :surprised

    What I wondering is how I can go about tackling this problem and actually somehow get a matrix from these conditions and whether anything I think I know about this problem are actually true or whether the complex numbers won't be conjugate because it is spiraling off axis and stuff like that... Any help would be greatly appreciated. :!!)
     
  2. jcsd
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