Question

How to find minimal polynomial usin characteristic polynomial?

Answer 1

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Two important facts relating the minimal and characteristic polynomials are

Both have exactly the same set of roots (in an algebraic closure of the ground field) namely that of the eigenvalues of the matrix. (Without going to the algebraic closure this means that they have the same set of irreducible polynomial factors, but in your case one of the polynomials, and therefore the other as well, is already split over Q.)The minimal polynomial divides the characteristic polynomial (this is the Cayley-Hamilton theorem).

So if you know the characteristic polynomial P, the minimal polynomial must be obtained by taking every distinct factor of P at least once, and at most as many times as it occurs as factor of P. Any polynomial so obtained (in your case there are 4 of them) can be the minimal polynomial.




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