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Hero
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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
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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malevolence19 Group TitleBest ResponseYou've already chosen the best response.0
\[\left[\begin{matrix}\psi & \xi \\ \chi & \zeta\end{matrix}\right] \implies (\psi  \lambda)(\zeta  \lambda)  \xi * \chi = 0\] 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

Hero Group TitleBest ResponseYou've already chosen the best response.1
Brilliantly done!
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
No, they are not the same always.
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
I need a counterexample.
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
Well, by default they wouldn't be the same matrices
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
\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

Hero Group TitleBest ResponseYou've already chosen the best response.1
The matrices that malevolence posted are not the same either are they?
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
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

Hero Group TitleBest ResponseYou've already chosen the best response.1
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

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
For example they cannot be similar either. Can you construct a counter example?
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
No, you cannot conclude that they have the same rank
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
\[ \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} \right)\\ and \\ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right) \] have the same eigenvalue but are not similar.
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
So what exactly am I supposed to conclude?
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
that they have the same eigenvalues ;)
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
The have the same determinant. They have the same characteristic polynomial. Something like that.
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
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

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
the product of the eigenvalues is always equal to the determinalt for any nxn matrix
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
@Zakron answered you correctly.
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
the trace will be the same
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
The trace is the sum of eigenvalues.
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
the trace is the sum of the main diagonal
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
but it is equal to the sum of the eigenvalues.
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
by theorem..not my definition
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
Of course.
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
that is what I was pointing out
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.2
@hero are you satisfied with what is posted above.
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
I can only award one medal
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.5
I think that if you are green (or purple) you should be able to give more than one medal.
 2 years ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.0
I don't think I can elaborate much on what Zarkon said do you have a specific question?
 2 years ago

Hero Group TitleBest ResponseYou've already chosen the best response.1
The question I had for you Turing had nothing to do with this question.
 2 years ago

malevolence19 Group TitleBest ResponseYou've already chosen the best response.0
I love this conversation haha @TuringTest @Zarkon @eliassaab
 2 years ago
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