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Homework Help: Eigenvalue and diagonalisation question

  1. Apr 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the eigenvalues and eigenvectors of the following matrix: M =
    1 1
    0 1
    Can this matrix be diagonalised?

    2. Relevant equations

    3. The attempt at a solution

    The characteristic equation is [itex](1 - \lambda)^{2} = 0[/itex] which gives [itex]\lambda = 1[/itex]. Substitute [itex]\lambda = 1[/itex] and eigenvector = {x,y} into the eigenvalue equation gives the two equations x+y = x and y = y. The first equation implies that y = 0. The second equation is redundant. So, x is free to assume any complex value. So, the eigenvalue is 1 and the eigenvector is {1,0}.

    I think everything I have done so far is fine. If it isn't, please point out.

    The problem starts with the second part: 'Can this matrix be diagonalised?' I know that to diagonalise a matrix is equivalent to changing the basis of the matrix and the eigenvectors. I am not quite sure I get this or the fact that the eigenvectors have to span the ??? to accomplish the diagonalisation.

    Thanks in advance for any help.
  2. jcsd
  3. Apr 3, 2012 #2


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    Homework Helper

    hi failexam! :smile:
    changing the basis doesn't change the behaviour …

    the space spanned by the eigenvectors of the original matrix is one-dimensional (the line y = o)

    how many dimensions is the space spanned by the eigenvectors of a diagonal matrix? :wink:

    (alternatively, if it was diagonalised, what would those diagonal entries be?)
  4. Apr 3, 2012 #3


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    Science Advisor

    An n by n matrix is "diagonalizable" if and only if it has n independent eigenvectors. If a matrix is NOT diagonalizable, it can be put in "Jordan Normal Form". And the matrix in this problem already is in Jordan Normal Form!
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