Find the minimal polynomial of $(x-1)^2(x-2)^3(x-3)$? Given ch. polynomial is $(x-1)^2(x-2)^3(x-3)$ and for the eigenvalue $1$, we have one eigenvector, for eigenvalue $2$ we have $2$ eigenvector and for $3$ we have $1$ eigenvector. Now determine the minimal polynomial?
I know that minimal polynomial divides the ch. polynomial, so I list the probable candidates as $$(x-1)(x-2)(x-3)$$ $$(x-1)(x-2)^2(x-3)$$ $$(x-1)^2(x-2)^2(x-3)$$ and $$(x-1)^2(x-2)(x-3).$$ But how to determine which one of these serves the purpose? There is no additional information given. Help me out

In fact there exist operators having your prescribed characteristic polynomial for each of the possible minimal polynomials provided:
Indeed, 


*

*for $(x−1)(x−2)(x−3)$ we have the diagonal matrix diag$(1,1,2,2,2,3)$

*for $(x−1)^2(x−2)^2(x−3)$; 
$$\left(\begin{smallmatrix}1 & 1& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 0 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
which is unique up to permutation of the Jordan blocks.

*for $(x−1)(x−2)^2(x−3)$; 
$$\left(\begin{smallmatrix}1 & 0& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 0 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
again unique up to permutation of the Jordan blocks.
and finally


*

*for $(x−1)^2(x−2)^3(x−3)$; 
$$\left(\begin{smallmatrix}1 & 1& 0& 0&0 &0\\  0& 1& 0& 0&0 &0 \\ 0& 0& 2& 1 & 0 & 0 \\0 & 0 & 0& 2 & 1 & 0 \\ 0 & 0 & 0 & 0& 2 & 0 \\ 0 & 0 & 0 & 0 &0 &3 \end{smallmatrix}\right)$$
up to permutation of the Jordan blocks.


Additionally, the combinatorial theory behind these ideas is very interesting. What you have specified in your problem (which is merely a characteristic polynomial) is 


*

*the dimension of the vector space (a.k.a. the size of the matrix in question); this is the degree of the characteristic polynomial.

*the algebraic multiplicity of each of the eigenvalues occurring in its upper triangular form (i.e. exactly which values will appear on the diagonal of the Jordan form)

*The different possible minimal polynomials hold the geometric multiplicity of each of the eigenvalues which specify the size of the maximum Jordan block for each of the eigenvalues in question.
