Proving Jordan Canonical Form for Similarity of Matrices with Same Polynomials

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To prove the similarity of matrices A and B with the same characteristic and minimal polynomials, it is essential to analyze their Jordan canonical forms. The key result states that if A and B are 3x3 nilpotent matrices, they are similar if they share the same minimal polynomial. Given that no eigenvalue has an algebraic multiplicity greater than 3, the structure of their Jordan blocks can be directly linked to their minimal polynomials. It is also crucial to confirm that the field F contains all roots of the characteristic polynomial, as this allows for the complete decomposition of the matrices into their Jordan forms. Thus, leveraging the relationship between the minimal polynomial and Jordan form will facilitate the proof of similarity for matrices A and B.
Bhatia
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I have to prove the following result:

Let A,B be two n×n matrices over the field F and A,B have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than 3, then A and B are similar.

I have to use the following result:

If A,B are two 3×3 nilpotent matrices, then A,B are similar if and only if they have same minimal polynomial.

Please suggest how to proceed.
 
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you need to understand the relationship between these polynomials and the jordan form. do you know how to prove the result you are allowed to use? Do you realize it is a special case of your problem? and are you allowed to assume that the field contains all roots of the characteristic polynomial?
 
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