1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding the eigenvectors for T()

  1. Dec 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Which of the following is not an eigenvector for [tex]
    T \left(
    \left[ {\begin{array}{cc}
    x \\
    y \\
    \end{array} } \right] \right) =
    \left[ {\begin{array}{cc}
    x + y \\
    x+ y \\
    \end{array} } \right]
    [/tex] ?

    A) v = [-2 -2]T
    B) v = [1 -1]T
    C) v = [1 2]T
    D) All are eigenvectors

    2. Relevant equations

    Ax = [tex]\lambda[/tex]x

    3. The attempt at a solution
    My problem is that all eigenvectors I've computed have come from 2x2 matrices. My best guess on starting is
    [tex]
    T \left(
    \left[ {\begin{array}{cc}
    -2 \\
    -2 \\
    \end{array} } \right] \right) =
    \left[ {\begin{array}{cc}
    -4 \\
    -4 \\
    \end{array} } \right]
    \left[ {\begin{array}{cc}
    -2 \\
    -2 \\
    \end{array} } \right]
    =
    [/tex]

    but this obviously doesn't work because of the size. How do I find an eigenvalue of a 2x1 matrix? Is it possible? Am I even looking at this correctly?

    Edit: I think I've figured it out. First, I constructed the standard matrix for T and got [tex]\left[ {\begin{array}{cc}
    1&1 \\
    1&1 \\
    \end{array} } \right] [/tex]
    Then I used Ax = [tex]\lambda[/tex]x with the vectors given to find the eigenvalues. Letter C didn't have an eigenvalue, so that is the answer.
    [tex]
    T \left(
    \left[ {\begin{array}{cc}
    -2 \\
    -2 \\
    \end{array} } \right] \right) =
    \left[ {\begin{array}{cc}
    1&1 \\
    1&1 \\
    \end{array} } \right]
    \left[ {\begin{array}{cc}
    -2 \\
    -2 \\
    \end{array} } \right] =
    \left[ {\begin{array}{cc}
    -4 \\
    -4 \\
    \end{array} } \right] = 2
    \left[ {\begin{array}{cc}
    -2 \\
    -2 \\
    \end{array} } \right]
    [/tex]
    [tex]
    T \left(
    \left[ {\begin{array}{cc}
    1 \\
    -1 \\
    \end{array} } \right] \right) =
    \left[ {\begin{array}{cc}
    1&1 \\
    1&1 \\
    \end{array} } \right]
    \left[ {\begin{array}{cc}
    1 \\
    -1 \\
    \end{array} } \right] =
    \left[ {\begin{array}{cc}
    0 \\
    0 \\
    \end{array} } \right] = 0
    \left[ {\begin{array}{cc}
    1 \\
    -1 \\
    \end{array} } \right]
    [/tex]
    [tex]
    T \left(
    \left[ {\begin{array}{cc}
    1 \\
    2 \\
    \end{array} } \right] \right) =
    \left[ {\begin{array}{cc}
    1&1 \\
    1&1 \\
    \end{array} } \right]
    \left[ {\begin{array}{cc}
    1 \\
    2 \\
    \end{array} } \right] =
    \left[ {\begin{array}{cc}
    3 \\
    3 \\
    \end{array} } \right] = ???
    [/tex]
     
    Last edited: Dec 17, 2009
  2. jcsd
  3. Dec 17, 2009 #2

    Mark44

    Staff: Mentor

    Your answer is correct, but you have made extra work for yourself in this problem. You didn't need to find a matrix representation for this transformation. All you needed to do was determine whether T(x) is equal to a scalar multiple of x.
    For example, T(-2, -2) = (-4, -4) = 2*(-2, -2).
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook