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Showing for which h a matrix is diagonalizable

  1. Oct 30, 2016 #1
    1. The problem statement, all variables and given/known data
    For what ##h## is the matrix ##\begin{bmatrix}1 & -h^2 & 2h \\ 0 & 2h & h \\ 0 & 0 & h^2 \end{bmatrix}## diagonalizable with real eigenvalues? (More than one may be correct)

    a) -2, b) -1, c) 0, d) 1, e) 2

    2. Relevant equations


    3. The attempt at a solution
    We already know the eigenvalues, since the matrix is upper triangular. How do we proceed? Do we just plug in the values of h and see if it is diagonalizable? It seems like that would take a very long time...
     
  2. jcsd
  3. Oct 30, 2016 #2

    andrewkirk

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    Yes, I would say the question is poorly expressed. I think what they meant to ask here is 'for what ##h## is the matrix diagonalisable over the reals?', which means that the change of basis matrix used must have only real entries.

    There is low-hanging fruit that enables determining whether some of a-e satisfy the requirement, based on the fact that:
    1. any diagonal matrix is diagonalisable
    2. any ##n\times n## matrix over field ##F## with ##n## distinct eigenvalues in ##F## is diagonalisable over ##F##.
    Having picked that fruit, you will have fewer of a-e left to try to work out whether they are diagonalisable using more advanced means. Have a go at that first part first.

    PS If you get stuck when you are up to the 'more advanced means' for the remaining cases, have a look in this wiki section about characterisation of diagonalisability.
     
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