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A,B similar <=> Rank(A) = Rank(B)?

  1. Jun 22, 2010 #1
    So I was wondering if "A and B are similar" is equivalent to "Rank(A) = Rank(B)".

    So obviously "=>" is always true, but I can't find any information on "<=". It seems logical, but I can't find a way to prove it. Also, even finding a counter-example doesn't seem easy, because then you'd have to prove there isn't any invertible matrix P so that [tex]P^{-1} A P = B[/tex], so I suppose a counter-example should be done with reductio ad absurdum, but nothing strikes me as an obvious example.

    Any help?

    Thank you,
    mr. vodka
     
  2. jcsd
  3. Jun 22, 2010 #2

    Office_Shredder

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    What matrices are similar to the identity matrix?
     
    Last edited: Jun 22, 2010
  4. Jun 22, 2010 #3
    Actually I'm not sure?

    But it gave me the idea that if a matrix is similar to the identity matrix, it has the same determinant, thus 1. Yet there are non-singular matrices with determinants not equal to one, thus giving me a reduction ad absurdum :) thus it's not an equivalency.

    Thank you.
     
  5. Jun 23, 2010 #4

    radou

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    Let A = I and B = λ I, so r(A) = r (B). Can A and B be similar?
     
  6. Jun 23, 2010 #5
    False even in the 1x1 case.
     
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