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Elementary Linear Algebra - Similar Matrices and Rank

  1. Dec 1, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose matrices A and B are similar. Explain why they have the same rank.



    2. Relevant equations



    3. The attempt at a solution
    So if A and B are similar, then there is some invertible matrix P such that B = P^-1AP. I have been trying to find some way to relate rank(A) to rank(P^-1AP) but I can't figure it out. I feel like maybe i'm missing some intuition about this. This comes at the end of a section about linear transformations.

    Thank you to anyone who can help.
     
  2. jcsd
  3. Dec 1, 2011 #2

    HallsofIvy

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    What is the definition of "rank"?
     
  4. Dec 1, 2011 #3
    "The dimension of the row (or column) space of a matrix is called the rank of A and is denoted by rank(A)."

    Thinking about this I recall that the rank of a linear transformation is equal to the rank of the matrix for T (where T is defined by T(x) = Ax). So if A and B both represent the same linear transformation but with respect to different bases, then they will have the same rank. This relationship is described by them being similar. P and P^-1 are the transition matrices between the bases.

    Is my thinking headed in the right direction? I still feel unclear about this. Is there a way to discuss this without reference to transformations and just from the definition of being similar?

    Thank you for your time.
     
  5. Dec 1, 2011 #4

    micromass

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    The rank of A is the dimension of the image of A. So take a basis of the image of A. Can you maje this into a basis of the image of [itex]PAP^{-1}[/itex]??
     
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