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Block matrix transformation of specific form

  1. Feb 20, 2013 #1
    Hi everyone,

    I am trying to solve the following problem. Is there exist a transformation matrix T, different then the block diagonal, with all blocks the same, such that the form of the matrix A=[A1 A2 ; I 0], is preserved? All blocks of A are in R^{nxn}, I is identity and 0 is zero matrix. In other words, is there exist matrix T (again, not block diagonal with all blocks the same) such that T^{-1}*A*T=B, where B=[B1 B2 ; I 0]?
    By intuition, such matrix T does not exist, but I do not know how this can be shown.
    If anyone has an idea about this please help. Thank you in advance.

    Nicolas
     
    Last edited: Feb 20, 2013
  2. jcsd
  3. Feb 20, 2013 #2

    mfb

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    So T should be like [t t; t t]?
    For the lower blocks of B, this would give the equations 1*t + 0*t = 1 and 1*t + 0*t = 0 => contradiction
     
  4. Feb 20, 2013 #3
    Thanks for your reply.

    Since I am looking for the existence of T, in general it could be
    T=[T11 T12 ; T21 T22]. I know that T=[T 0; 0 T] holds, but I want to show that this is the only case. For example T=[T11 0 ; 0 T22] cannot hold, neither the example that you proposed.
     
  5. Feb 20, 2013 #4

    mfb

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    Ok, I was not sure what "with all blocks the same" means.
    Anyway, [t t; t t] matrix would not have an inverse matrix.
     
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