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2x2 matrix A has only one eigenvalue λ with eigenvector v

  1. Jan 31, 2010 #1
    This is a revision problem I have come across,

    I have completed the first few parts of it, but this is the last section and it seems entirely unrelated to the rest of the problem, and I can't get my head around it!

    Suppose that the 2x2 matrix A has only one eigenvalue λ with eigenvector v, and that w is a non zero vector which is not an eigenvector..show that:

    a) v and w are linearly independent
    b) the matrix with respect to the basis {v, w} is
    (λ c
    0 λ)
    for some c =not to 0
    c) for a suitable choice of w, c = 1

    I am stuck.
    I know how to show that the eigenvalues are linearly independent, but how do I show that these two vectors are linearly independent to eachother?
    as for b and c i dont know where to start! Please help!
    Last edited: Jan 31, 2010
  2. jcsd
  3. Jan 31, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper

    If v and w are linearly dependent, then w is a multiple of v, so obviously w is also an eigenvector.

    Get some sleep! :zzz:​
  4. Jan 31, 2010 #3
    Re: Eigenvalues/vectors

    ahh ok..so I can prove by contradiction! thank you that helps massively for part a!
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