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Matrix Eigen Vector Question

  1. Apr 6, 2010 #1
    1. The problem statement, all variables and given/known data

    I've done part A, and part D is easy. I'm stuck with part B. I have no idea what a "right hand side vector" is...

    HtQcr.png

    2. Relevant equations



    3. The attempt at a solution

    Part A: All eigenvectors are valid. Eigenvalues are 1, 1/2 and 1/3.
     
  2. jcsd
  3. Apr 6, 2010 #2

    Mark44

    Staff: Mentor

    Write each of the vectors e1, e2, and e3 as a linear combination of k1, k2, and k3.

    The idea is to write Ane1 as An(c1k1 + c2k2 + c3k3).

     
  4. Apr 6, 2010 #3
    How do I find the constants and the e vectors? I don't think I really understand what what I'm supposed to do represents...
     
  5. Apr 6, 2010 #4

    Mark44

    Staff: Mentor

    The ei are just the standard basis vectors for R3.
    e1 = <1, 0, 0>T
    e2 = <0, 1, 0>T
    e3 = <0, 0, 1>T

    You must have been taught how to write a vector (such as k1) as a linear combination of other vectors. Check your book and/or notes.
     
  6. Apr 6, 2010 #5
    I looked but it doesn't really say. We don't have a textbook for this actually and the notes are very vague. This is what it says:

    We can write
    x(0) = c1 k1 + c2 k2 + c3 k3 + c4 k4 (6.2)
    for some coefficients c1 , c2 , c3 and c4 uniquely determined. Equation (6.2) can
    be written in matrix-vector form
    T c = x(0) (6.3)
    where c = (c1 , c2 , c3 , c4 )T and T is the 4
    × 4 matrix with eigenvectors k1 , k2 , k3
    and k4 in its columns. Solving (6.3) (I used MATLAB) gives c1 = 1/2, c2 = 1/2,
    c3
    ≈ −0.6830 and c4 ≈ 0.1830. With these values of c (6.2) is a representation
    of x(0) as a linear combination of eigenvectors of P .
     
  7. Apr 6, 2010 #6

    Mark44

    Staff: Mentor

    Put this in terms of what you have. You want to find constants c1, c2, c3 so that
    e1 = c1 k1 + c2 k2 + c3 k3

    and the same thing (with different sets of constants) for e2 and e3.

    Then you can calculate Ane1 = An(c1 k1 + c2 k2 + c3 k3).
    This works out to c1 An k1 + c2 An k2 + c3 An k3.

    Since the ki's are eigenvectors, you can calculate A ki, and hopefully you have learned something about the eigenvalues of An.
     
  8. Apr 7, 2010 #7
    Ok like this?:

    e1 = k1 - k2 + k3
    e2 = 2 k1 + 0 k2 + k3
    e3 = 0 1 0

    Now what? I think I need to understand what's going on, not just what I need to do...
     
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