Similar matrix and characteristic polynomial

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Discussion Overview

The discussion revolves around the relationship between similar matrices and their characteristic polynomials, specifically addressing the assertion that while similar matrices share the same characteristic polynomial, the reverse is not necessarily true. Participants explore how to demonstrate this concept through examples and proofs.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that two similar matrices have the same characteristic polynomial, but two matrices with the same characteristic polynomial need not be similar.
  • Another participant suggests finding two matrices that are not similar yet share the same characteristic polynomial as a proof strategy.
  • A participant proposes aiming for an easy characteristic polynomial to facilitate the proof.
  • One participant recommends using the difference of squares, specifically the polynomial x^2-1, as a potential characteristic polynomial.
  • Another participant notes that if two matrices have the same characteristic polynomial, they must have the same eigenvalues but may have different eigenvectors, suggesting that one could be diagonalizable while the other is not.
  • A participant mentions that to prove the similarity condition, one can use the change of basis approach, indicating that matrices A and B are similar if there exists an invertible matrix Q such that B=Q^-1*A*Q.
  • Another participant hints at using a 2x2 Jordan block and a 2x2 identity matrix as examples to illustrate the concept.

Areas of Agreement / Disagreement

Participants generally agree on the foundational statements regarding similar matrices and characteristic polynomials, but the discussion remains unresolved regarding specific examples and proofs to illustrate the concepts.

Contextual Notes

There are limitations in the discussion regarding the assumptions needed for the examples and proofs, as well as the dependence on definitions of similarity and characteristic polynomials. Some mathematical steps remain unresolved.

Eric Nelson
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A true statement: Two similar matrices have the same characteristic polynomial.

The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

HOw can I prove this?

Any help appreciated.
 
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it's enough to find two matrices which are not similar yet have the same char poly
 
Note: Pick an easy charisteristic polynomial to aim for.
 
can you suggest an easy characteristic polynomial, I'm stumped on this concept. thank you
 
You can try a difference of squares. For instance, x^2-1 Also, you can think of what you can do to a matrix that wouldn't change the characteristic polynomial.
 
If two matrices have the same characteristice polynomial, the obviously they have the same eigenvalues. What you want is that they have different eigenvectors. You can do that is one is diagonalizable but the other isn't. To give an example in 2 by 2 matrices, they must have only a single eigenvalue. On matrix would then have 2 \independent eigenvectors, the other only 1 eigenvalue.
 
to prove the part that is true, use change of basis. matrices A and B are similar if there exists an invertible Q such that B=Q^-1*A*Q. But all you're really doing is putting your linear transformation into another basis, so they'll have the same characteristic polynomial. You can check this by showing the change of basis, then taking the determinant of A and B.
 
Take a 2x2 jordan block and also a 2x2 identity matrix, and...
 

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