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Complex and upper triangular matrices

  1. Oct 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that for all complex two by two matrices, they will be upper triangular matrices (edit: i think what is meant by upper triangular matrices is that PAP^-1 will be upper triangular matrices - the wording of the question i was given was a little misleading it seems)

    2. Relevant equations

    A=PDP^-1

    3. The attempt at a solution

    The way I tried to do it was to show that A will have n distinct eigenval, whereby I let

    A=[a b
    c d]

    i found the eigenvals for that. then and for the case a=d, tried to show that the eigenvec are not linearly independent. However, this is the correct way to do it? or is there a much easier solution to this question?
     
    Last edited: Oct 10, 2009
  2. jcsd
  3. Oct 10, 2009 #2
    Possibly interesting:
    An endomorphism is a upper triangularizable endomorphism if and only if it does cancel a split polynomn(in your case, in C[X]).
     
  4. Oct 10, 2009 #3
    oh dear. i don't even know what an endomorphism is. i probably can't use it to solve my problem!
     
  5. Oct 10, 2009 #4
    In the context of linear algebra, 'endomorphism' is just the proper (and short) way of saying 'linear transformation of a space into itself'.


    At any rate, I don't understand the question. An upper triangular matrix is nilpotent and certainly not all linear transformations of C2 are nilpotent.
     
  6. Oct 10, 2009 #5

    I'm not quite sure what you meant by nilpotent either (do you mean equal to zero when raised to a certain power?)

    By I think what the question means is that there will be an invertible matrix P s.t. PAP^-1 is upper triangular, and is something to do with diagonalisation. not sure though.
     
  7. Oct 11, 2009 #6
    An upper triangular matrix is not always nilpotent (a "strict" upper triangular one is).
    And the link between endomorphism and your problem is that every square matrix can be interpreted as the matrix of an endomorphism in different basis.
    But basically, you don't need to introduce it in your problem. You just need to find out a split polynomn in C[X] that cancles your matrix ( I think you can find one of degree 2 or 3).
     
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