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Linear Algebra - Showing a Matrix is not Diagonalizable

  1. Dec 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that the matrix A is not diagonalizable. Explain your reasoning.

    A=\begin{bmatrix}k&0\\0&k\\\end{bmatrix}

    2. Relevant equations
    Important theorem: A nxn matrix is diagonalizable if and only if it has n linearly independent eigenvectors.


    3. The attempt at a solution
    Since A is triangular, the eigenvalues are the entries on the main diagonal. In this case the only eigenvalue is k. So then I solve for B = (λI - A) where λ=k, which turns out to equal a 2x2 zero matrix. Then I solve Bx = 0 to try and find the eigenvectors. Here is where I think i'm going wrong. Since B is the zero matrix, I believe that x1 = t, x2 = s, where t and s are any real number. So I find that the vector x is equal to t(1,0) + s(0,1) which would indicate to me that the matrix has two linearly independent eigenvectors and should be diagonalizable.

    But this is the opposite of which I wished to prove! So clearly my thinking must be wrong. The solution for this problem says that a basis for the eigenspace is simply {(0,0)}, so since A does not have two linearly independent eigenvectors, it does not satisfy the theorem I have above and cannot be diagonalizable.

    My problem is that I don't see how you can say from Bx=0 where B is the 2x2 zero matrix that x can only be (0,0) and thus the basis for the eigenspace is only {(0,0)}. Can't x be any vector?

    Thank you.

    EDIT: Another point. Isn't this matrix also symmetric (A=Atranspose)? Then shouldn't it be diagonalizable (it's already diagonal anyways)?
     
    Last edited: Dec 12, 2011
  2. jcsd
  3. Dec 12, 2011 #2

    lanedance

    User Avatar
    Homework Helper

    I would go with your EDIT, this is already a diagonal matrix.

    Are you sure the matrix is written correctly?
     
  4. Dec 12, 2011 #3
    Thank you for replying. Yes, it is written correctly.

    There must be some idea I'm missing here in solving the problem. I have trouble believing the book is incorrect on this, since I'm usually wrong on such things.
     
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