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I was wondering what is the relationship between minimal polynomial and the Jordan Canonical Form. Before given a matrix, all you need is to compute the characteristic polynomial to determine the Jordan form and using the dot diagram(a la Friedberg) it is unique. Thus I am wondering what additional information does minimal polynomial give.

For example the question:

Find all possible $7$ by $7$ Jordan Canonical matrix with characteristic polynomial $t^2(2-t)^3(3-t)(4-t)$ and the minimal is the same as the characteristic polynomial.

Solved:

Chapter 7.3 Problem 13 from Friedberg.

If $T \in L(V)$ and char(T) splits. Let $λ_1, λ_2, … , λ_k$ be distinct eigenvalues of $T$ and for each $i$ let $p_i$ be the order of the largest Jordan block corresponding to $λ_i$ in a Jordan canonical form of $T$. Then the minimal polynomial of $T$ is $(t-λ_1)^{p1}(t-λ2)^{p1} \cdots (t-λ_k)^{pk}$

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