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

[nonequilibrium]

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 P^{-1} A P = B, 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

 

[Office_Shredder]

What matrices are similar to the identity matrix?

 

[nonequilibrium]

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.

 

[radou]

Let A = I and B = λ I, so r(A) = r (B). Can A and B be similar?

 

[g_edgar]

False even in the 1x1 case.