Finding Jordan forms over the complex numbers

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To compute Jordan forms over the complex numbers from a minimal polynomial, it is essential to consider the nature of the roots. If the roots are real, the Jordan forms can still be constructed over the complex field, as complex numbers include real numbers. The structure of Jordan forms remains consistent, with eigenvalues on the diagonal and either "0" or "1" above them. Even with real eigenvalues, the Jordan form can still be represented in the complex number system. Understanding this allows for accurate computation of Jordan forms regardless of the eigenvalue nature.
chuckles1176
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So I am trying to compute all possible Jordan forms over the complex numbers given a minimal polynomial. My question is this: If the roots of the minimal polynomial are both real, should I proceed as if all of the possible forms are over real numbers?
 
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The Jordan forms are always matrices with the eigenvalues on the diagonal and either "0" or "1" (in some notations, below) above each eigenvalue. If all the eigenvalues are real, I can see no way to introduce complex numbers into them.
 
chuckles, do you by any chance go to Michigan State?
 

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