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Hero Group Title

If a 2x2 matrix has the same two eigenvalues as another 2x2 matrix, what conclusions can you infer about the two matrices? Make some hypotheses and attempt to prove them, or refute them via counterexamples.

  • 2 years ago
  • 2 years ago

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  1. malevolence19 Group Title
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    \[\left[\begin{matrix}\psi & \xi \\ \chi & \zeta\end{matrix}\right] \implies (\psi - \lambda)(\zeta - \lambda) - \xi * \chi = 0\] Take the transpose of that matrix you have: \[\left[\begin{matrix}\psi & \xi \\ \chi & \zeta\end{matrix}\right]^T=\left[\begin{matrix}\psi & \chi \\ \xi & \zeta\end{matrix}\right] \implies (\psi - \lambda)(\zeta - \lambda) - \xi * \chi = 0\] They are the same. I'm not sure if you can ALWAYS assume this but it seems pretty logical. I also don't know any theorems to name.

    • 2 years ago
  2. Hero Group Title
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    Brilliantly done!

    • 2 years ago
  3. eliassaab Group Title
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    No, they are not the same always.

    • 2 years ago
  4. Hero Group Title
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    Hmmm

    • 2 years ago
  5. Hero Group Title
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    I need a counterexample.

    • 2 years ago
  6. Hero Group Title
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    Well, by default they wouldn't be the same matrices

    • 2 years ago
  7. eliassaab Group Title
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    \left( \begin{array}{cc} 2 & 4 \\ 0 & 3 \\ \end{array} \right) \\ \left( \begin{array}{cc} 2 & 0 \\ 0 &3 \\ \end{array} \right) Have the same eigenvalues 2 and 3 but they are not the same.

    • 2 years ago
  8. Hero Group Title
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    The matrices that malevolence posted are not the same either are they?

    • 2 years ago
  9. eliassaab Group Title
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    Ok. I thought he wanted to conclude that if two matrices have the same eigenvalues then they are the same. This is not true in general. What do you think the answer to your question is?

    • 2 years ago
  10. Hero Group Title
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    Well, apparently, there's more than one conclusion. If you have two matrices that are not the same but have the same eigenvalues, I'm not really sure what to conclude which is why I posted the question.

    • 2 years ago
  11. eliassaab Group Title
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    For example they cannot be similar either. Can you construct a counter example?

    • 2 years ago
  12. Zarkon Group Title
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    same rank

    • 2 years ago
  13. eliassaab Group Title
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    No, you cannot conclude that they have the same rank

    • 2 years ago
  14. eliassaab Group Title
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    \[ \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} \right)\\ and \\ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right) \] have the same eigenvalue but are not similar.

    • 2 years ago
  15. Hero Group Title
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    So what exactly am I supposed to conclude?

    • 2 years ago
  16. Zarkon Group Title
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    that they have the same eigenvalues ;)

    • 2 years ago
  17. eliassaab Group Title
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    The have the same determinant. They have the same characteristic polynomial. Something like that.

    • 2 years ago
  18. Hero Group Title
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    So any two matrices with the same eigenvalues will have the same determinant and characteristic polynomial? That doesn't seem accurate. Maybe for certain cases, but not for every case

    • 2 years ago
  19. Zarkon Group Title
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    the product of the eigenvalues is always equal to the determinalt for any nxn matrix

    • 2 years ago
  20. eliassaab Group Title
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    @Zakron answered you correctly.

    • 2 years ago
  21. Zarkon Group Title
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    the trace will be the same

    • 2 years ago
  22. eliassaab Group Title
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    The trace is the sum of eigenvalues.

    • 2 years ago
  23. Zarkon Group Title
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    the trace is the sum of the main diagonal

    • 2 years ago
  24. eliassaab Group Title
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    but it is equal to the sum of the eigenvalues.

    • 2 years ago
  25. Zarkon Group Title
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    by theorem..not my definition

    • 2 years ago
  26. eliassaab Group Title
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    Of course.

    • 2 years ago
  27. Zarkon Group Title
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    that is what I was pointing out

    • 2 years ago
  28. eliassaab Group Title
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    I see/

    • 2 years ago
  29. eliassaab Group Title
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    @hero are you satisfied with what is posted above.

    • 2 years ago
  30. Hero Group Title
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    yes,thanks

    • 2 years ago
  31. eliassaab Group Title
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    yw

    • 2 years ago
  32. Hero Group Title
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    I can only award one medal

    • 2 years ago
  33. Zarkon Group Title
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    I think that if you are green (or purple) you should be able to give more than one medal.

    • 2 years ago
  34. Hero Group Title
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    I agree

    • 2 years ago
  35. TuringTest Group Title
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    I don't think I can elaborate much on what Zarkon said do you have a specific question?

    • 2 years ago
  36. Hero Group Title
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    The question I had for you Turing had nothing to do with this question.

    • 2 years ago
  37. malevolence19 Group Title
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    I love this conversation haha @TuringTest @Zarkon @eliassaab

    • 2 years ago
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