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

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Discussion Overview

The discussion revolves around the relationship between matrix similarity and rank, specifically whether "A and B are similar" is equivalent to "Rank(A) = Rank(B)". Participants explore implications, counter-examples, and specific cases related to this concept.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant, mr. vodka, suggests that while "Rank(A) = Rank(B)" implies "A and B are similar", they are uncertain about the reverse implication and seek a proof or counter-example.
  • Another participant questions what matrices are similar to the identity matrix, hinting at the properties of determinants in relation to similarity.
  • A different participant proposes that if a matrix is similar to the identity matrix, it must have the same determinant, which leads them to conclude that not all non-singular matrices can be similar to the identity, thus arguing against the equivalency.
  • One participant provides a specific example with matrices A and B, where A is the identity matrix and B is a scalar multiple of the identity, questioning their similarity despite having the same rank.
  • A later reply asserts that the claim is false even in the simplest case of 1x1 matrices.

Areas of Agreement / Disagreement

Participants express differing views on the equivalency of matrix similarity and rank, with no consensus reached. Some argue for the implications of rank on similarity, while others provide counter-examples that challenge this notion.

Contextual Notes

The discussion highlights the complexity of proving relationships between matrix properties, with participants acknowledging the difficulty in finding counter-examples or proofs. There are unresolved assumptions regarding the definitions of similarity and rank.

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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What matrices are similar to the identity matrix?
 
Last edited:
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.
 
Let A = I and B = λ I, so r(A) = r (B). Can A and B be similar?
 
False even in the 1x1 case.
 

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