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

  • Thread starter nonequilibrium
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In summary, the conversation discusses the equivalence of "A and B are similar" and "Rank(A) = Rank(B)", with the question of whether "<=" is always true. The speaker also mentions using reductio ad absurdum to find a counter-example. The conversation also touches on the idea of matrices similar to the identity matrix and their determinants. The conclusion is that A and B cannot always be similar, as shown in the example of A = I and B = λ I.
  • #1
nonequilibrium
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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
 
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  • #2
What matrices are similar to the identity matrix?
 
Last edited:
  • #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.
 
  • #4
Let A = I and B = λ I, so r(A) = r (B). Can A and B be similar?
 
  • #5
False even in the 1x1 case.
 

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

What does it mean for two matrices to be similar?

Two matrices are considered similar if they have the same size and shape, and their corresponding entries share the same relationships. This means that for matrices A and B to be similar, there must exist a matrix P such that A = PBP-1.

How can you tell if two matrices are similar?

To determine if two matrices are similar, you can compare their ranks. If the rank of matrix A is equal to the rank of matrix B, then A and B are considered similar. This is known as the Rank-Nullity Theorem.

3. What is the significance of two matrices being similar?

When two matrices are similar, it means that they represent the same linear transformation in different coordinate systems. This allows us to use different matrices to represent the same transformation, which can be useful in certain mathematical calculations.

4. Can two matrices be similar if they have different dimensions?

No, two matrices cannot be similar if they have different dimensions. As mentioned before, for matrices to be similar, they must have the same size and shape. This includes having the same number of rows and columns.

5. Is similarity an equivalence relation?

Yes, similarity is an equivalence relation. This means that it is reflexive, symmetric, and transitive. In other words, a matrix is always similar to itself, if A is similar to B then B is also similar to A, and if A is similar to B and B is similar to C, then A is also similar to C.

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