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Homework Help: Triangularity of a matrix over the unitarian space

  1. Jul 2, 2011 #1
    In my book I have a theorem that every matrix is similar to a triangular matrix if the characteristic polynomial could be wrote as a multiplication of linear terms.
    Now suppose we are dealing with a matrix over a unitarian space we know that we can write its characteristic polynomial as a multiplication of linear forms, now what I'm trying to prove is that I can find an orthogonal/orthonormal basis in which the matrix is triangular.

    Is this true?
    Can you give me a hint how to prove such as thing?
    [If this is true many examples that I see in my book could be proved in more compact and neat form, so I think this is very useful theorem...]

    [EDIT] I have an idea now to use the fact that every matrix in the unitarian space similar to a jordan matrix, so I only need to show that jordan basis could be transformed into orthonormal jordan basis...
    Last edited: Jul 2, 2011
  2. jcsd
  3. Jul 2, 2011 #2
    Hi estro! :smile:

    Firstly, what is the "unitarian space", I have never heard of this term before...

    Secondly, (depending on what unitarian space is), I don't think your theorem is true. If there were an orthogonal basis in which the matrix is triangular, then this would mean that there is a orthogonal basis of (generalized) eigenvectors. But this is not true in general.
  4. Jul 2, 2011 #3
    unitarian space is a liner space V over the field C where the Inner product defined.
    I'll think about what you said.
    Last edited: Jul 2, 2011
  5. Jul 2, 2011 #4
    Ah, that would be a unitary space. I did see that term once, I completely forgot it :frown:
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