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

  • #1
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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
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
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What matrices are similar to the identity matrix?
 
Last edited:
  • #3
1,434
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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.
 
  • #4
radou
Homework Helper
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Let A = I and B = λ I, so r(A) = r (B). Can A and B be similar?
 
  • #5
607
0
False even in the 1x1 case.
 

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