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Minimal polynomial and diagonalization of a block matrix

  1. May 31, 2014 #1
    The problem statement, all variables and given/known data.

    Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##.

    The attempt at a solution.

    I am a pretty lost with this problem: I know that, thinking in terms of columns ##X## is the set of matrices with just one linearly independent column. In an ##n\times n## matrix there are ##n## columns, so I thought that maybe there could be ##n## representatives of this equivalence relation, but I couldn't prove it and in fact I am not at all convinced this is true. I would appreciate suggestions to solve the problem.
     
  2. jcsd
  3. May 31, 2014 #2

    AlephZero

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    That is correct, but rank-1 matrices have other properties that could be more useful to answer the question. Reading http://en.wikipedia.org/wiki/Matrix_similarity should give you some ideas.

    It's not true. There are an infinite number of equivalence classes, and an infinite number of matrices in each class.
     
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