3x3 similar matrices defined by characteristic and minimal polynomials

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SUMMARY

Two 3x3 matrices are similar if and only if their characteristic polynomial and minimal polynomial are equal. This property holds true due to the limited ways to decompose 3x3 matrices into Jordan blocks, which describe matrices up to conjugacy. In contrast, this relationship does not extend to 4x4 matrices, where additional complexity arises. The discussion emphasizes the significance of Jordan blocks in understanding matrix similarity.

PREREQUISITES
  • Understanding of Jordan blocks in linear algebra
  • Familiarity with characteristic and minimal polynomials
  • Knowledge of matrix similarity and conjugacy
  • Basic concepts of 3x3 matrix decomposition
NEXT STEPS
  • Study the properties of Jordan blocks in detail
  • Explore the differences between 3x3 and 4x4 matrix similarity
  • Learn about the implications of minimal polynomials in matrix theory
  • Investigate applications of matrix similarity in linear transformations
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Students and professionals in mathematics, particularly those focusing on linear algebra, matrix theory, and advanced topics in eigenvalues and eigenvectors.

JamesTheBond
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Why do you guys think that given two 3x3 matrices, they are similar if and only if their characteristic polynomial and minimal polynomial are equal (this reasonably fails for 4v4 matrices though)?
 
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Just consider the Jordan Blocks
 
Not exactly sure what you mean. How do Jordan blocks get involved?
 
Jordan blocks are what describe matrices up to conjugacy. In a 3x3 matrix there are very few ways to decompose as Jordan block matrices, which answers your question as to why 3x3 (and 2x2) matrices are completely determined by their minimal polynomials.
 

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