3x3 similar matrices defined by characteristic and minimal polynomials

[JamesTheBond]

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)?

 

[matt grime]

Just consider the Jordan Blocks

 

[JamesTheBond]

Not exactly sure what you mean. How do Jordan blocks get involved?

 

[matt grime]

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.