Finding Jordan forms over the complex numbers

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SUMMARY

This discussion focuses on computing Jordan forms over the complex numbers based on a given minimal polynomial. It establishes that if the roots of the minimal polynomial are real, one should not assume that the Jordan forms will be restricted to real numbers. The Jordan forms consist of matrices with eigenvalues on the diagonal and either "0" or "1" above each eigenvalue, indicating the presence of generalized eigenvectors. The conversation highlights the importance of recognizing the nature of eigenvalues when determining the appropriate Jordan forms.

PREREQUISITES
  • Understanding of Jordan forms in linear algebra
  • Familiarity with minimal polynomials and their properties
  • Knowledge of eigenvalues and eigenvectors
  • Basic concepts of complex numbers in mathematics
NEXT STEPS
  • Research the computation of Jordan forms for matrices with complex eigenvalues
  • Study the implications of generalized eigenvectors in Jordan form representation
  • Explore the relationship between minimal polynomials and Jordan forms
  • Learn about the application of Jordan forms in solving differential equations
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Mathematicians, students of linear algebra, and anyone interested in advanced topics related to Jordan forms and eigenvalue analysis.

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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